Actual Infinity in Reality?

[polymath257]

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One of the issues here, I think, is that there are two definitions of the term 'infinite' at play. I think there are good reasons for rejecting one of them and accepting the other.

Definition 1: Something is infinite if it is unbounded.

Definition 2: Something is infinite if it cannot be put into a correspondence with some counting number.

Confusion between these definitions is at least part of the problem here. So, let's discuss the relative merits of the two: The first assumes a *process* that doesn't stop while the second assumes a completed object that doesn't correspond with some counting number.

So, for example, the sequence

1,2,3,4,5,6

is both a process that stops and it can be put into correspondence with a counting number ( 6 ), so it satisfies both definitions.

But there is a problem with definition 1. It is ambiguous. The question of whether something is 'bounded' depends on the order properties of that object: which things are larger and which are smaller than others. if two *different* notions of order give different answers, the definition needs to take that into account.

So, to do this, we need to say that something is unbounded *in a particular order*. So, for example, the list of objects

.9
.99
.999
.9999
.99999
...

is an infinite collection of objects (whether you want to say potential or actual is irrelevant here). Each one of the sequence has another that is 'more'. So, with the *internal* ordering from the sequence itself, it is 'unbounded'. But, in the order provided by *all* numbers, this sequence *is* bounded: every single thing in that sequence is smaller than 1.

So, if we are using definition 1, is this sequence unbounded (because for each term there is one larger), or is it bounded (because every term is smaller than 1)? The definition is ambiguous.

On the other hand, using the *second* definition, this sequence is infinite: there is no counting number that is in correspondence with it. NOTHING about process or order or anything else is required to get this answer. And the answer is unambiguous.

So, this makes the second definition 'infinitely' preferable. The ambiguity of the term 'bounded' means that many situations will give ambiguous answers. Furthermore, an attention to the discussion of Zeno's paradox shows that exactly this ambiguity is active in the argument. The above sequence is 'both' unbounded and 'bounded'. That is regarded as a contradiction, when it is actually just a shift in definitions of 'bounded'. No contradiction follows.

Next, the emphasis on process in the first definition forces finitistic reasoning: it is 'begging the question'. Instead, considering relevant sets to exist and be complete allows deeper and and more informative analysis. So, for example, we can consider the set

N={x: x is a counting number}

This is a set: we can tell exactly when something is a member and exactly when something is not a member. There is no ambiguity in the membership in this collection. Hence, it is reasonable to consider this a *completed* collection. There is no 'process' involved, just unambiguous questions about membership. And, it is clear, for example, that 82793 is a member and 34.95723 is not. Simple.

Now, a lot of hay has been made about the assumption in mathematics called the 'axiom of infinity', where it is assumed that a completed infinity actually exists. Some claim this to be 'begging the question'. Let's analyze this claim.

Suppose that actual infinities are, in fact, contradictory. Then adding this axiom would inevitably lead to contradictions. In other words, there would be statements *that are well formed* where both p and (not p) can be proven. It is a fact that no such statements have ever been produced.

But we can go farther. If there are no contradictions actually produced, we can safely and without fear of contradiction add this axiom to our system!

Why would we want to do so? Well, first of all, because it leads to simpler proofs of results that are awkward to prove otherwise. Second, because it leads to deeper insights. And third, because it leads to a more precise and symmetric system of thought.

First, even basic facts about the counting numbers, which *can* be proved without the assumption of infinite sets have *much* simpler proofs with that assumption. Many times, even the statement of fundamental ideas (such as mathematical induction) are ponderous and forced in a finitistic system, but natural and easy when infinities are assumed.

Next, the assumption of infinities leads to deeper insights into even finite sets. By having an extra layer of analysis opened up, we find that even statements about *finite* sets can be stated easier proved quicker. By having the infinite to work with, the finite comes into clearer view.

And, finally, having the assumption of infinite sets allows a MUCH more symmetric and complete system of thought. Most of mathematics over the last century and a half has been based on the use of infinite sets and the subject has expanded greatly since that point. As Hilbert said: nobody will remove us from the paradise Cantor opened up to us.

Next, the question of applicability of mathematics to the real world.

Mathematics is a *language*. When and if it is applicable to the real world is a matter of observation and testing. it is something to be decided upon by the scientists, not the philosophers (who have been so wrong so often). If infinite space and time is a *useful* for our investigation of the universe (and they are), then those ideas have to be taken seriously, in spite of the thrashing of obsolete philosophies.

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I would agree, if they are the same, then they are equal.... that there is no space between them could be confusing.

Well, that was clearly the meaning of what was said: zero distance, which implies equality.