As I see it, you are trying to make a proof by contradiction here by postulating an infinite object and trying to show that its existence must results in an impossibility. But the contradictions you describe don’t follow from your argument, in my mind. So…

I think the first problem with the logic is here – I’d agree that an object must have a non-zero, positive length to exist (physically), but it doesn’t follow that if the length of an object is undefined that it cannot exist. In other words, to exist, the object must have length, but it is not a requirement for existence that the length be measurable or defined.devans99 wrote: ↑June 26th, 2020, 7:11 amA finite brick with left and right end is the type of brick we are used to. An actually infinite length brick would have a left end but continue forever to the right - IE it would have no right end - if it had a right end it would of course be a finite length brick.

If we reduce the length of our actually infinite brick to finite length, then we have a finite length brick with a left end but no right end. But such a brick cannot possibly exist - if it has no right end, it has no middle (because the middle would count as the right end) and if it has no right end, it cannot have a left end either (as the left end would also count as the right end) - so the brick cannot exist.

Reenforcing this, a brick with a left end but no right end obviously has an UNDEFINED length and all objects must have a non-zero, positive length to exist.

And as you said a little further along that:

I would again agree that an infinite object cannot be measured, but that an object

*has*a length has nothing to do with whether or not you have ability to measure it. So again, a brick with no end cannot be measured, which means only that the length cannot be determined, not that it cannot exist.

I’m not clear what you mean about a physical object having the ‘entire structure of the natural numbers’, but I think you’re continuing the same error here. Natural numbers are counts, not measures. In other words, there’s a difference between infinite in quantity and infinite in size. You’re applying the logic of numbers to measurements but they’re not the same thing. One is analytic and one is synthetic/empirical.devans99 wrote: ↑June 26th, 2020, 7:11 amReturning to the ‘set’ of natural numbers:

{1, 2, 3, 4, 5, 6, 7, 8, … }

If something existed in reality with the entire structure of the natural numbers, then like the brick, it would have no right end. If it has no right end, then it has no end-1 (because that would count as the end). If it has no end-1, it cannot have an end-2. If it has no end-n, it has no end-(n+1). We can then use mathematical induction to work backwards through all the natural numbers to show the sequence has no start either. So anything with the structure of the entire natural numbers (IE actual infinity) cannot possibly exist in reality, or even logically - it would have no end; therefore no start and so it would not exist at all.

So (actual) infinity appears logically impossible?

Lastly, you’ve said a couple of times that:

...and I’m not a mathematician by any means, but I'd just point out that I don’t think the proofs say that infinite

*objects*are possible - they say that infinite

*sets*are possible. So similarly, sets have a quantity but do not have a dimension that can be measured, so the analogy cannot work to disprove the possibility of an object of infinite length. As I see it, that's the essential confusion here, and the problem with the concept of infinity in general, that it's problematic to use mathematical concepts that are analytical to draw conclusions that rely on empirical observations for verification since these are two different realms of knowledge.

A proof by contradiction succeeds if it is one that results in a

*logical*impossibility, but you are trying to show proof by describing a situation that is an

*empirical*impossibility. But empirical observations can't be said to be either possible or impossible - we just observe what we are able to observe. So if you postulate that an infinite brick exists as a premise, then you can't rely on the limitation of our ability or to observe it, or to imagine that we could potentially observe it, to prove the impossibility of its existence.

Thoughts?