rationalskepticism.org

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?