Wortfish wrote:

Newmark wrote:
...and the inverse of a number that is between 0 and 1 lies in what range? Compare this with your statement that "the inverse of all irrational numbers [...] lie within 0 and 1". And thank you for proving yourself wrong again!

The inverse of an irrational number....not the irrational number itself lies within 0 and 1. Stop twisting what I wrote.

In the post that I quoted, you gave an example of an irrational number that lies between 0 and 1, that is sqrt(2) -1. What is 1/(sqrt(2) -1)? What is -(sqrt(2) -1)? Just own up to that "the inverse of all irrational numbers [...] lie within 0 and 1" was wrong, and I'll quit bugging you about it. But if you want to keep painting yourself into a corner, go right ahead.

I'm terribly sorry, but I just can't find facepalm picture big enough. You don't really have any clue about what you are talking about, do you? That you are making things up about how you think mathematics work would be cute if I had any reason to assume that you where willing to learn from your errors. Instead, you blindly dismiss well-established mathematical facts as "illusions", which quite frankly is a downright pathetic argument in a mathematical discussion.

I am not making up anything. Amazingly, you fail to realise that taking the inverse of an increasingly large range of unbounded integers (scaled by a factor of 10) generates a (near) infinite contiunum of real numbers between 0 and 0.5. That is why it is an illusion to suppose that there is an infinity of real numbers bounded between any two integers.

What I don't fail to realize is that what you describe only describes rational numbers. Not any irrational numbers, neither algebraic nor transcendental. Mathematically speaking, the rationals are not an continuum. The rationals are countably infinite (just as the integers), a continuum (such as the reals) is uncountably infinite. Mathematical proofs of these fact do exist. I gave you a constructive proof below about how you can construct reals that can't be rationals below. As I've said, you are not contributing anything new that mathematicians haven't already thought of. Read up on the field, or embarrass yourself further; your choice.

For the record, Cantor proved that the reals were uncountably infinite in the late 19th century. Your "objections" only represents an incredibly naive version of set theory, and such issues are address if you actually take the time to properly learn anything about the subject.

Not all real numbers lie between 0 and 1.

...and you think this addresses what, exactly? Cantor certainly didn't claim that; if I've mistakenly made that claim anywhere, I immediately concede that I was wrong. But please do continue: how do you think that this fact invalidates Cantor's proof?

And I notice that you dodge a particular one of my questions: now that you've dismissed (among other things) set theory and calculus as "illusion" without "application to reality", what do you think of any technology that is in any way based on them?

I never said set theory has no applications. I said it has no application whatsoever to the problem of an infinite past.

Which makes make curious about what branch of mathematics you think can model time in any meaningful way. Care to elaborate? And that still leaves open the question as to what you want to replace set theory and calculus with, since you've dismissed central parts of both as "illusion"...

I note how you totally failed to address my objection to your forever-moving object. Care to address it?

Sure. Just copy/paste or link your favorite version of that argument (you've posted several), and I'll answer it. Greyman and Thomas has already pointed out most of your fallacies though, but I might be able to squeeze a few moer inconsistencies out of it.

Newmark wrote:
In the post that I quoted, you gave an example of an irrational number that lies between 0 and 1, that is sqrt(2) -1. What is 1/(sqrt(2) -1)? What is -(sqrt(2) -1)? Just own up to that "the inverse of all irrational numbers [...] lie within 0 and 1" was wrong, and I'll quit bugging you about it. But if you want to keep painting yourself into a corner, go right ahead.

I don't even understand what your objection amouts to. The sqrt(2) -1 already lies between 0 and 1 and so there is no need to take its inverse. However, the sqrt(2) does not lie within that range, but its inverse does. So, either all positive irrational numbers exist between 0 and 1 or their inverses do. That was always my argument.

What I don't fail to realize is that what you describe only describes rational numbers. Not any irrational numbers, neither algebraic nor transcendental. Mathematically speaking, the rationals are not an continuum. The rationals are countably infinite (just as the integers), a continuum (such as the reals) is uncountably infinite. Mathematical proofs of these fact do exist. I gave you a constructive proof below about how you can construct reals that can't be rationals below. As I've said, you are not contributing anything new that mathematicians haven't already thought of. Read up on the field, or embarrass yourself further; your choice.

Who cares? A rational approximation to an irrational number is virtually identical so long as you use big enough integers. Do you not think that an infinitely-sided polygon is, for all intents and purposes, a perfect circle?

...and you think this addresses what, exactly? Cantor certainly didn't claim that; if I've mistakenly made that claim anywhere, I immediately concede that I was wrong. But please do continue: how do you think that this fact invalidates Cantor's proof?

And I have proven that all real numbers, including irrational ones, are just scaled inverses of integers approaching infinity. I can easily estimate PI to 1000 places precisely by using the division of two big enough integers. It may not be accurate to the 1001st place, so I would then need to use even bigger numbers.

Which makes make curious about what branch of mathematics you think can model time in any meaningful way. Care to elaborate? And that still leaves open the question as to what you want to replace set theory and calculus with, since you've dismissed central parts of both as "illusion"...

The illusion is in supposing that potential infinities are actual infinities.

Sure. Just copy/paste or link your favorite version of that argument (you've posted several), and I'll answer it. Greyman and Thomas has already pointed out most of your fallacies though, but I might be able to squeeze a few moer inconsistencies out of it.

You were the one who brought up the object that has always been moving. I responded in detail but your never replied. My objection was basically that something that has been moving forever has already traversed all possible points and so, as paradoxical and counter-intuitive as it may appear, cannot proceed any further.

Thomas Eshuis wrote:You have failed to demonstrate infinity to be a problem or impossible.

He can't make a case for that, because it would make his believed-in god untenable.

Wortfish wrote:You're missing the point entirely. If I have a big enough integer I can use its inverse, multiplied by a factor of 10, to get to the approximation of any irrational number that is only infinitesimally inaccurate.

Apologies if it sounds like I'm defending Wortfish here. I only respond in case I manage to say something interesting, which is all cranks are good for.

Computers generally only have a certain number of digits (or bits) to work with for calculations (at least if they want to be fast). And so they can't efficiently represent arbitrary irrational numbers. The standard way to approximate them in hardware is basically by using the binary version of scientific notation, which is called "floating point." But this itself relies on having specialised hardware for dealing with numbers expressed in scientific notation.

Back in the early days of home computers (and this includes the first Playstation), you didn't generally have this specialised hardware, so another way to approximate irrational numbers was to use fixed rather than floating point. This effectively meant approximating your irrationals by integers, exactly as Wortfish describes above: pick a suitable factor of 10 (or 2, since it's a computer), and then declare that all your integers are actually that factor bigger than the number you intend.

This causes some issues, because, if you're working at a factor of, say, 8, then when you multiply, you're now at a factor of 16, and so you have to shave off 8 of your digits, quite possibly losing precision in the process. You can avoid this by doubling the number of bits you use to hold the product, so that you don't lose the information, but it was still less than ideal.

archibald wrote:

Thomas Eshuis wrote:You have failed to demonstrate infinity to be a problem or impossible.

He can't make a case for that, because it would make his believed-in god untenable.

Actually, Newark's obsession with infinite sets is his own undoing: http://milesmathis.com/cant.html

You cannot add one to infinity, because an infinite set is a complete set. An infinite set is complete in the fullest sense, meaning that there exists nothing outside the set. If you have an infinite set of people, then all people are in that set.

Hence, if something has always been moving, it has covered all possible ground with an infinite number of moves.

From the link:

This is intended as proof that the set of irrationals is larger than the set of integers, as well as that the set of irrationals is not countable.

But we cannot possibly "count" irrationals in order, like we do integers. Notice that there is no way you could ever make a chart like this in the first place, since you could never choose a first irrational after zero. That first irrational in the chart has an infinite number of a's in it. And, whichever "first irrational" you choose, I can choose a smaller one.

I'll just leave this here for Greyman, scott and Newark to laugh at.

If anything, Wortfish's posts are good examples of Dunning-Kruger in action.

VazScep wrote:From the link:

This is intended as proof that the set of irrationals is larger than the set of integers, as well as that the set of irrationals is not countable.

But we cannot possibly "count" irrationals in order, like we do integers. Notice that there is no way you could ever make a chart like this in the first place, since you could never choose a first irrational after zero. That first irrational in the chart has an infinite number of a's in it. And, whichever "first irrational" you choose, I can choose a smaller one.

I'll just leave this here for Greyman, scott and Newark to laugh at.

This Miles Mathis character has also beaten established physics in provided us with a Grand Unification Theory, enlightened us about how the Boston bombing was fake, and proved his mathematical genius by finally establishing that π is 4. Needless to say, mainstream science hasn't caught up to his level of "genius" yet...

Sorry....how does all this point to there being a big giant who lives in the sky again? I must have missed something...like facts.

Wortfish wrote:

Newmark wrote:
In the post that I quoted, you gave an example of an irrational number that lies between 0 and 1, that is sqrt(2) -1. What is 1/(sqrt(2) -1)? What is -(sqrt(2) -1)? Just own up to that "the inverse of all irrational numbers [...] lie within 0 and 1" was wrong, and I'll quit bugging you about it. But if you want to keep painting yourself into a corner, go right ahead.

I don't even understand what your objection amouts to. The sqrt(2) -1 already lies between 0 and 1 and so there is no need to take its inverse. However, the sqrt(2) does not lie within that range, but its inverse does. So, either all positive irrational numbers exist between 0 and 1 or their inverses do. That was always my argument.

Those were not the qualifications you specified in the statement from you I quoted above, which says something else entirely. You should really use more precise language to avoid this sort of confusion. This leaves only the question as to why you think the wholly uncontroversial statement "either all positive irrational numbers exist between 0 and 1 or their inverses do" provides any form of interesting argument for or against anything...

What I don't fail to realize is that what you describe only describes rational numbers. Not any irrational numbers, neither algebraic nor transcendental. Mathematically speaking, the rationals are not an continuum. The rationals are countably infinite (just as the integers), a continuum (such as the reals) is uncountably infinite. Mathematical proofs of these fact do exist. I gave you a constructive proof below about how you can construct reals that can't be rationals below. As I've said, you are not contributing anything new that mathematicians haven't already thought of. Read up on the field, or embarrass yourself further; your choice.

Who cares? A rational approximation to an irrational number is virtually identical so long as you use big enough integers. Do you not think that an infinitely-sided polygon is, for all intents and purposes, a perfect circle?

Mathematicians care. Virtually identical =/= identical. Mathematically speaking, there is a world of difference between the set of rationals and the set of reals. Approximations do have real world applications (in fact, fields such as numerical analysis are wholly devoted to approximations), but that does not in any way invalidate any other field of mathematics. You may not care, but if you really intend to discuss what is and is not logically possibly, your position is rubbish if you cannot justify it with regard to relevant fields.

...and you think this addresses what, exactly? Cantor certainly didn't claim that; if I've mistakenly made that claim anywhere, I immediately concede that I was wrong. But please do continue: how do you think that this fact invalidates Cantor's proof?

And I have proven that all real numbers, including irrational ones, are just scaled inverses of integers approaching infinity.

No, you have not proven that all. That you think that you have done that is an indication that you quite frankly lack to competence to see why you are wrong. I can do no further for you until you drop your hubristic notions about disproving well-established mathematical fields which you know little to nothing about, and pick up a text book on basic set theory and actually learn something.

I can easily estimate PI to 1000 places precisely by using the division of two big enough integers. It may not be accurate to the 1001st place, so I would then need to use even bigger numbers.

You can also approximate PI to 4, just as your favorite "mathematician" does. But an approximation is not the same as the real value. That π cannot be accurately written as a fraction has been proved multiple times.

Which makes make curious about what branch of mathematics you think can model time in any meaningful way. Care to elaborate? And that still leaves open the question as to what you want to replace set theory and calculus with, since you've dismissed central parts of both as "illusion"...

The illusion is in supposing that potential infinities are actual infinities.

Then give me proper definitions of those two terms. In particular, how they relate to various mathematical concepts, such as Dedekind-infinite sets and their cardinalities, and limits. Otherwise, the supposed "illusions" is nothing else than your failure to grasp the relevant definitions.

And you have still not provided us with a coherent model of time that does not rely on set theory.

Sure. Just copy/paste or link your favorite version of that argument (you've posted several), and I'll answer it. Greyman and Thomas has already pointed out most of your fallacies though, but I might be able to squeeze a few moer inconsistencies out of it.

You were the one who brought up the object that has always been moving. I responded in detail but your never replied. My objection was basically that something that has been moving forever has already traversed all possible points and so, as paradoxical and counter-intuitive as it may appear, cannot proceed any further.

Well, since you won't give me a reference to what specific iteration of your argument you want me to address, I'll just pick one that seems to fit best (even if it was an answer to Thomas, not me). If you think I'm addressing the wrong one, it is up to you to supply a correct version. Anyway, let's get started:

Wortfish wrote:1. You suggest that if something has always been moving , it doesn't follow that it hasn't reached every point.

That is indeed correct. Infinite sets doesn't need to include every possible element. The negative integers don't include all the integers, the integers don't include all the reals, the reals don't include all the complex numbers, etc. It is also quite possible to have an set that excludes specific elements, e.g. ℤ \ {0} (all the integers except 0), which is still a set of (countably) infinite size. More importantly, it is entirely mathematically possible for an infinite set to have an upper and/or lower bound, e.g. the negative numbers.

I do, quite frankly, have no idea about what your favorite "mathematician" wibbles on about here, because by any reasonable definition of "complete", his assertions are either trivially wrong or utterly misleading.

2. So, you think it can be true that it has reached point A, but not point B further along the line.

Indeed. If we have reached a specific point, we have an infinity with an upper bound. If you happen to have a "definition" (i use the term very loosely in this case) of infinity that does not allow for upper bounds, e.g. Wortfish-kind infinities, then that definition is simply not applicable to the past. You can't argue against something that doesn't apply to the situation (that would be a strawman), so you need to either based your objections on the model proposed or supply an adequate reason as to why that particular model doesn't apply, of which you have done none.

If I have been counting negative integers up until this point, I have not yet reached zero.

3. This would mean that not enough time has elapsed for it to move to point B.

Since when? Every point in the past (if we model it as the negative integers) is a finite distance in the past. That means that from any point X in the past, a finite amount of time has elapsed when we reach point A, which is indeed not enough time to reach point B.

The only relevant point to measure from that you can mean would be a "beginning" (in this case, a lower bound). To measure from an undefined point that is not part of the model can tell you absolutely nothing about the distance between any elements in that model. That you insist on doing this (even though I've pointed it out multiple times) is rather dishonest at this point.

Alternatively, you are conflating set size with the relations between the elements in it again, which is simply trivially wrong.

4. But that is ridiculous given that it has had forever to get not just to A but also to B.

Define "forever" as you would use it in the context of a past. If this definition includes "endless", it does not apply to the past, and you should use a term that actually applies to the model being discussed instead (e.g. "beginningless").

In the given model, that is (as I have shown) certainly not "ridiculous".

5. As such, we must conclude that something always moving cannot be at a point for which there are subsequent points.

6. Therefore, if something is at point A, but is yet to reach point B, it must have not always been moving.

7. So, anything currently moving in a straight trajectory must have begun to move at some point.

If we were talking about Wortfish-infinities, you might have had a point. We are not, so all these conclusion doesn't follow from the premises. You are wrong.

In short, for your argument to work, you either need shoehorn in a model that doesn't apply (e.g. Wortfish-kind infinities), or you need to introduce a "beginning" in a model that doesn't need one. Mind you, I am only talking about models based on integers in this post; divisibility would be one about limits.


And while we're talking baout beginnings, let's have a side-order of that too:

Wortfish wrote:

Newmark wrote:
And around in circles we go. You still haven't explained why it can't just have been going forever, thus not having the need of "get going". Hint: Craig "supports" his claim that everything must have a beginning by asserting that infinities are impossible...

If I never began my journey to Shangri-la, then I will never reach Shangri-la, or indeed any place. So it is with time. If time has no beginning, there is no journey or flow of time as we ourselves experience.

Insofar that this is an "explanation" for why everything must have a beginning, it relies on your argument above that constant velocity is impossible, which in turn relies on the need to have a beginning. A circular argument is circular.

But this analogy also has quite a problem when you try apply it to time. You see, all that is observed to be required for a moment to flow to the next is the existence of those particular moments. This is literally what we observe all the time: one moment flowing into the next. What we don't observe is moments popping into existence out of nowhere all over the place*, which is very much unlike journeys WITHIN time, which we do indeed observe starting quite frequently. If you want this analogy to hold, show me a finite interval of moments that has a beginning (as in, the first movement has no moment preceding it) and an end (no moment follows the last one)...

* By this, I don't mean to discount any cosmologies that proposes that time "began" (for some relevant definition of the word) at some point, e.g. the big bang. But, as far as I know, they deduce this from whatever model they're working from in conjunction with what the available evidence tells us, not from some esoteric notions on the nature of time based on incoherent definitions of "infinity". As I've said, you are perfectly free to (try to) base your argument on cosmology instead, but those who know something about cosmology around here may have a thing or two to say about it.

Newmark wrote:
Those were not the qualifications you specified in the statement from you I quoted above, which says something else entirely. You should really use more precise language to avoid this sort of confusion. This leaves only the question as to why you think the wholly uncontroversial statement "either all positive irrational numbers exist between 0 and 1 or their inverses do" provides any form of interesting argument for or against anything...

Of course, I was not referring to negative irrational numbers.

Mathematicians care. Virtually identical =/= identical. Mathematically speaking, there is a world of difference between the set of rationals and the set of reals. Approximations do have real world applications (in fact, fields such as numerical analysis are wholly devoted to approximations), but that does not in any way invalidate any other field of mathematics. You may not care, but if you really intend to discuss what is and is not logically possibly, your position is rubbish if you cannot justify it with regard to relevant fields.

Nobody doubts that irrational and transcendental numbers are unique, and cannot be expressed either arithmetically and/or algebraically. But that doesn't mean they cannot be rationally approximated using big integers.

No, you have not proven that all. That you think that you have done that is an indication that you quite frankly lack to competence to see why you are wrong. I can do no further for you until you drop your hubristic notions about disproving well-established mathematical fields which you know little to nothing about, and pick up a text book on basic set theory and actually learn something.

Look, all an irrational real number is is a very large integer with a decimal point somewhere. This may horrify you, but that's the reality. This makes a nonsense of the claim that there is an infinity of real numbers bounded between 0 and 1.

Then give me proper definitions of those two terms. In particular, how they relate to various mathematical concepts, such as Dedekind-infinite sets and their cardinalities, and limits. Otherwise, the supposed "illusions" is nothing else than your failure to grasp the relevant definitions.

Potential infinites consist of endlessly dividing something into endlessly small components. The illusion of supposing there is an actual infinity of real numbers between 0 and 0.5 is that all you are doing is taking the inverse of ever bigger integers to get such a range. Actual infinites do not exist because, if they did, it would not be possible to add to them. From your knowledge of set theory, you should know that an infinite set is a complete set. You cannot add anything to a complete set.

That is indeed correct. Infinite sets doesn't need to include every possible element. The negative integers don't include all the integers, the integers don't include all the reals, the reals don't include all the complex numbers, etc. It is also quite possible to have an set that excludes specific elements, e.g. Z \ {0} (all the integers except 0), which is still a set of (countably) infinite size. More importantly, it is entirely mathematically possible for an infinite set to have an upper and/or lower bound, e.g. the negative numbers.

Not sure about that. If an infinite set does not include every possible element, it is not complete.

Since when? Every point in the past (if we model it as the negative integers) is a finite distance in the past. That means that from any point X in the past, a finite amount of time has elapsed when we reach point A, which is indeed not enough time to reach point B.

A finite distance, yes, from the present. But that only works if the present has been reached which is not possible if it takes forever - an infinite number of moments - to reach it.

The only relevant point to measure from that you can mean would be a "beginning" (in this case, a lower bound). To measure from an undefined point that is not part of the model can tell you absolutely nothing about the distance between any elements in that model. That you insist on doing this (even though I've pointed it out multiple times) is rather dishonest at this point. Alternatively, you are conflating set size with the relations between the elements in it again, which is simply trivially wrong.

If there is no beginning, then there is no point for which the object has not reached.

If we were talking about Wortfish-infinities, you might have had a point. We are not, so all these conclusion doesn't follow from the premises. You are wrong. n short, for your argument to work, you either need shoehorn in a model that doesn't apply (e.g. Wortfish-kind infinities), or you need to introduce a "beginning" in a model that doesn't need one. Mind you, I am only talking about models based on integers in this post; divisibility would be one about limits.

Recall that infinite sets are complete sets. They contain all possible elements or else they would not be infinite. Likwise, an object that has moved an infinite distance has moved all possible distances and, therefore, cannot go any further.

But this analogy also has quite a problem when you try apply it to time. You see, all that is observed to be required for a moment to flow to the next is the existence of those particular moments. This is literally what we observe all the time: one moment flowing into the next. What we don't observe is moments popping into existence out of nowhere all over the place*, which is very much unlike journeys WITHIN time, which we do indeed observe starting quite frequently. If you want this analogy to hold, show me a finite interval of moments that has a beginning (as in, the first movement has no moment preceding it) and an end (no moment follows the last one)...

We observe the flow of time precisely because there is a beginning from which all time flows. If we don't have a beginning, there can be no passage of time since all possible moments necessarily have already endured. Think of a lack of a beginning as an infinite black hole and not, as you do, something that is a finite distance from the present (or whatever you mean).

Wortfish wrote:
This makes a nonsense of the claim that there is an infinity of real numbers bounded between 0 and 1

What is the first real number after zero? It would be zero followed by an infinity of zero after the decimal point with one
added on to the end. But you cannot add anything onto infinity because it is never ending. Therefore the number of reals
there are between zero and one has to be infinite. And there are an infinity of numbers between any two reals no matter
how close they actually are. That is because many reals will themselves extend to infinity either as integers or irrationals

surreptitious57 wrote:But you cannot add anything onto infinity because it is never ending.

Oh, you totally can. This is what the ordinals are all about. "Ordinal" isn't a fancy maths word. It just refers to a use of numbers to order things: first rather than one. Second rather than two. Third rather than three. One hundredth rather than one hundred.

It's perfectly acceptable to have infinitieth, and, more, the idea of infinitieth is almost demanded when one starts thinking seriously about ordering. The infinitieth number is that object that you append to the end of the counting numbers, normally written ω. It is defined as an extra object such that all finite numbers are stipulated to be smaller than it.

With ω, you could imagine a decimal number where there are an infinite number of digits after the decimal point, plus one extra at the ω place.

Naturally, you don't have to stop at ω. There is ω + 1, defined as an extra object such that all finite numbers and ω are stipulated to be smaller than it.

Of course, there is ω + 2, ω + 3, and so on, and when you hit the limit there, you introduce ω + ω. This game goes on for a long time, even if you only introduce a countable number of these new objects. Then things get weird.

There's an axiom you can assert which says that the real numbers can be enumerated by ordinals. There, you will have a first real, a second real, a third real, and so on, up to the ωth real, and then the (ω+1) real, and so on, up to the (ω + ω) real, and so on, up to the ordinal 2^ω (two to the power of ω). The ordering implied here isn't the usual ordering of the reals, of course.

I had heard of the ordinals before though I did not know what their actual function was
I still find it very counter intuitive that something can be added on to infinity but I shall
have to suspend disbelief and simply accept it from now on given that it is actually true

surreptitious57 wrote:I had heard of the ordinals before though I did not know what their actual function was
I still find it very counter intuitive that something can be added on to infinity but I shall
have to suspend disbelief and simply accept it from now on given that it is actually true

I suppose it's counterintuitive if you think about having to traverse an infinity so that you can put another object on the end. I'd say that idea is pretty weird. But that's not really the way you think about it. When it comes to ordinals, all you are really thinking about is how you answer the question "is x less than y?"

For counting numbers, we all know how to answer this question. 1 is less than 2. 100 is less than 1000. And so on. If you throw an extra object into this domain, call it whatever you like, say "Bob", you now have to add in the rules that allow you to answer "is Bob less than y" and "is x less than Bob."

To get a number that's bigger than all the naturals, the rule couldn't be simpler:

Bob is not less than anything.
All natural numbers are less than Bob.

Done. We have an object that's bigger than all natural numbers. That's literally all that's going on here. For good measure, he's the working computer code:

Code: Select all

-- Extend a domain with a Bob
data WithBob a = A a | Bob deriving Eq

-- In an ordered domain, everything is less than or equal to Bob.
instance Ord a => Ord (WithBob a) where
  A x <= A y = x <= y
  x <= Bob   = True


That explains it better and so makes me less likely to think of it as counter intuitive. Although force of habit means
it will take a while yet for me to stop thinking nothing can be added onto infinity for I always think the number line
extends to infinity in both directions. But now I shall have to remember that it can have an ordinal at each end also

You can even stack whole copies of the natural numbers on each end, so it looks like:

...,1000,...,100,...1,...,1000,...,100,...,1,...,1000,...,100,...,1,...1000,...,100,...1,...

In some sense, this is even simpler. We'll consider just the case that we have an infinite stack of copies of the natural numbers going in one direction. That is, there is a first stack of natural numbers, then a second stack, then a third, and so on. So to point to an actual element in this infinitely infinite structure, we need to know the stack number, and then we need to know the actual number inside that stack. In other words, each object is referenced by a pair (s,n). Here, s is the stack number, and n the element in the stack.

Comparison is then defined as:

(s1,n1) < (s2,n2) if:

s1 < s2
OR
s1 = s2 and n1 < n2.

You could extend this to triples, if you wanted to have stacks of stacks of stacks of natural numbers, and then to quadruples, and then you could extend it further to infinite lists of natural numbers. The orderings you define in this way are precisely the way that dictionaries order words: if the first letter is smaller, the word is smaller. Otherwise, you check if the second letter is smaller. And if necessary, you check the third letter, and so on. Hence, this is sometimes called dictionary ordering or lexicographic ordering. It's also the way you count up to ω2, ω3, and so on, and if you go with lists, you're counting up to ωω (not the same meaning as I had above with 2ω, which was an abuse of notation).

You can also get these stacks going in the other direction. Just have an extra component which is either L or R, where L < R, and make this the first component in your dictionary ordering.

Wortfish wrote:

Newmark wrote:
That is indeed correct. Infinite sets doesn't need to include every possible element. The negative integers don't include all the integers, the integers don't include all the reals, the reals don't include all the complex numbers, etc. It is also quite possible to have an set that excludes specific elements, e.g. Z \ {0} (all the integers except 0), which is still a set of (countably) infinite size. More importantly, it is entirely mathematically possible for an infinite set to have an upper and/or lower bound, e.g. the negative numbers.

Not sure about that. If an infinite set does not include every possible element, it is not complete.

Well, that is not quite how completeness is defined in set theory, but roughly yes.

However, a infinite set does not have to be complete ... or include "every possible element", for that matter.

For instance, the set of rational numbers is not complete, but it is still countable infinite. There is not a finite count of real numbers that can be exactly represented as the ratio of two integers, but neither can, as one of so many examples, √2 be exactly represented as the ratio of two integers.

(Yes, Wortfish, you can approximate it; but you have to approximate it, because you cannot be exact.)

Here's a situation that bothers me about infinity.

Take a point a distance of 1 unit from the origin. This is going to be the first point in a figure that we're going to build up. To build the rest of the figure, take an angle that is some irrational number of degrees, say the square root of 2 degrees. To create a second point, rotate the first point about the origin by that angle. To create the third point, rotate again. To create the fourth, rotate again. And so on, and so on.

This gives you a potential infinity of points, since rotating by a multiple of an irrational number of degrees can never take you back to a point you've already been to. Take, as our figure, all the points that can be generated by this method.

Now rotate the whole figure by that same angle. If you think about this, you realise that the first point rotates to the second, the second to the third, the third to the fourth, and so on. The figure maps to itself, expect for the very first point. That doesn't get mapped to.

So this means that the simple rotation deletes a point. In fact, you can delete any number of points you like by using the appropriate rotation. But if you rotate back in the other direction, you magically get all these points back. Rotation somehow generates matter out of nothing, and erases it just as easily.

If you use that axiom I mention above, you can play this game in three dimensions using a simple observation from group theory, and, with a few rotations and translations, you find that you can conjure up whole solid balls out of nothing. In particular, you can take a single solid unit ball and rotate bits of it into two solid unit balls.

I don't much like this.

Wortfish wrote:

Newmark wrote:Mathematicians care. Virtually identical =/= identical. Mathematically speaking, there is a world of difference between the set of rationals and the set of reals. Approximations do have real world applications (in fact, fields such as numerical analysis are wholly devoted to approximations), but that does not in any way invalidate any other field of mathematics. You may not care, but if you really intend to discuss what is and is not logically possibly, your position is rubbish if you cannot justify it with regard to relevant fields.

Nobody doubts that irrational and transcendental numbers are unique, and cannot be expressed either arithmetically and/or algebraically. But that doesn't mean they cannot be rationally approximated using big integers.

Approximations are not the same things as the value they approximate. Do learn this basic fact.

No, you have not proven that all. That you think that you have done that is an indication that you quite frankly lack to competence to see why you are wrong. I can do no further for you until you drop your hubristic notions about disproving well-established mathematical fields which you know little to nothing about, and pick up a text book on basic set theory and actually learn something.

Look, all an irrational real number is is a very large integer with a decimal point somewhere. This may horrify you, but that's the reality. This makes a nonsense of the claim that there is an infinity of real numbers bounded between 0 and 1.

Then you can tell me what integer that is equal to a (integer) multiple of π. Pro tip: since all integers are finite, they all have a finite length. Guess how many decimals π has...

Then give me proper definitions of those two terms. In particular, how they relate to various mathematical concepts, such as Dedekind-infinite sets and their cardinalities, and limits. Otherwise, the supposed "illusions" is nothing else than your failure to grasp the relevant definitions.

Potential infinites consist of endlessly dividing something into endlessly small components. The illusion of supposing there is an actual infinity of real numbers between 0 and 0.5 is that all you are doing is taking the inverse of ever bigger integers to get such a range.

So you randomly declare any realm of mathematics that you don't understand "an illusion". How very persuasive of you. Not that you cover the reals with this definition (since you don't understand what a real numbers is, given your above statement), but since it even by your definition exists just as many rationals between 0 and 1 as there are integers, I'm guessing that the integers would also be potentially infinite?

Actual infinites do not exist because, if they did, it would not be possible to add to them. From your knowledge of set theory, you should know that an infinite set is a complete set. You cannot add anything to a complete set.

Then you can define "complete set" for me, because I don't know of any such term that makes sense in this context. Sure, there are terms such as Dedekind completeness and Cauchy completeness, but far from all infinities are "complete" in that sense.

Now, I know that you are just quoting your favorite "mathematician", but if you want your opinions to be taken seriously, don't rely on the "knowledge" of someone who thinks π = 4...

That is indeed correct. Infinite sets doesn't need to include every possible element. The negative integers don't include all the integers, the integers don't include all the reals, the reals don't include all the complex numbers, etc. It is also quite possible to have an set that excludes specific elements, e.g. Z \ {0} (all the integers except 0), which is still a set of (countably) infinite size. More importantly, it is entirely mathematically possible for an infinite set to have an upper and/or lower bound, e.g. the negative numbers.

Not sure about that. If an infinite set does not include every possible element, it is not complete.

Indeed. That set is infinite, yet is not "complete" in the way you think that word means. As usual, you don't understand the words you use, and thus cannot provide any relevant definition for this usage.

Since when? Every point in the past (if we model it as the negative integers) is a finite distance in the past. That means that from any point X in the past, a finite amount of time has elapsed when we reach point A, which is indeed not enough time to reach point B.

A finite distance, yes, from the present. But that only works if the present has been reached which is not possible if it takes forever - an infinite number of moments - to reach it.

No. You haven't got a working model of infinity, because you don't understand the required mathematics. In a proper model (infinite or not), any moment can be reached be reached from any other moment in a finite number of steps, and a specific elements is reached from a given element in exactly one specific amount of steps. This has been explained to you. "Infinity" does not measure the distance between any two points, it can only be used to describe the size of the sets that represents the past, the future, or time as a whole. Again, set size =/= distance between elements.

And since you don't understand this model, you - again - attack a strawman of it by attempting to include a "beginning" to measure infinity from. Attack my model on on how it is actually stated, or show me why it isn't applicable (and no, all your hand waving so far has done no such thing); or simply continue to flaunt your ignorance.

The only relevant point to measure from that you can mean would be a "beginning" (in this case, a lower bound). To measure from an undefined point that is not part of the model can tell you absolutely nothing about the distance between any elements in that model. That you insist on doing this (even though I've pointed it out multiple times) is rather dishonest at this point. Alternatively, you are conflating set size with the relations between the elements in it again, which is simply trivially wrong.

If there is no beginning, then there is no point for which the object has not reached.

We've been over this. Do you think that the sets {..., -2, -1} and {..., -2, -1, 0} are equal? Both have no lower bound, and thus no beginning as I have defined it. You are, as usual, dead wrong.

If we were talking about Wortfish-infinities, you might have had a point. We are not, so all these conclusion doesn't follow from the premises. You are wrong. n short, for your argument to work, you either need shoehorn in a model that doesn't apply (e.g. Wortfish-kind infinities), or you need to introduce a "beginning" in a model that doesn't need one. Mind you, I am only talking about models based on integers in this post; divisibility would be one about limits.

Recall that infinite sets are complete sets allegedly something that I don't understand. They contain all possible elements or else they would not be infinite. Likwise, an object that has moved an infinite distance has moved all possible distances and, therefore, cannot go any further.

FIFY

You conception of what constitutes "all possible elements" is ill-defined (must the natural numbers contain every possible complex number?), and so is your idea of infinity (since you by your definition above can't tell me that the set of every integer except zero is infinite). In short, if you want to actually understand mathematics, don't take all your lessons from a conspiracy crackpot.

But this analogy also has quite a problem when you try apply it to time. You see, all that is observed to be required for a moment to flow to the next is the existence of those particular moments. This is literally what we observe all the time: one moment flowing into the next. What we don't observe is moments popping into existence out of nowhere all over the place*, which is very much unlike journeys WITHIN time, which we do indeed observe starting quite frequently. If you want this analogy to hold, show me a finite interval of moments that has a beginning (as in, the first movement has no moment preceding it) and an end (no moment follows the last one)...

We observe the flow of time precisely because there is a beginning from which all time flows. If we don't have a beginning, there can be no passage of time since all possible moments necessarily have already endured. Think of a lack of a beginning as an infinite black hole and not, as you do, something that is a finite distance from the present (or whatever you mean).

And here (aside from previously refuted points) you provide an ample example of why your concepts of infinity doesn't apply to the past, while at the same time displaying that you don't understand a model as simple as the negative numbers, and thus continue to argue against your strawman.

Yet again, "there must have been a beginning, because infinity is impossible, because there must have been a beginning"... Don't you every get dizzy?