Actual Infinity in Reality?

[polymath257]

Since there are new people joining the discussion, here is a summary of problems as I see it. 

INFINITE SET THEORY
Some have suggested that the area of infinite set theory in mathematics is an indication that an actual infinite is possible. So this is how that argument goes.

1. An actual infinite consists of real (not abstract) objects.
2. The Axiom of Infinity assumes an abstract object that is an actual infinite set
3. Paradoxes, contradictions, and absurdities (such as below) can be dismissed because we can apply infinite set theory parameters/restrictions to the actual objects
     Hilbert's Hotel
     Galileo's paradox
     Ross–Littlewood paradox
     Thomson's Lamp paradox
     Zeno's Paradoxes
4. Therefore actual infinities are possible.

Since (2) is of the class of axioms that are not self-evident, they are assumptions on which further mathematical equations can be developed (useful in calculus for example). To be clear, this axiom is not reasoned into--it is just assumed as a foundation for the subset of infinite set theory in mathematics. See this earlier post. 

This is CLEARLY a question-begging argument and therefore invalid. 

On the contrary, the fact that 1) these ideas are useful (and required) for calculus, 2) that calculus has served as a fundamental tool to help us understand reality, and 3) the fact that no contradiction is produced from these assumptions
is quite sufficient to show the *possibility* of an actual infinity.

Most importantly, the lack of an actual contradiction (as opposed to simply having different rules) is quite enough to make these ideas a possibility. That they are actually used and helpful is what makes them true in the real world.

But, let's go back a bit. What are the 'self-evident' axioms? Previous posts have suggested 1. Existence 2.Consciousness 3. Law of Identity 4. Law of Non-Contradiction, and 5. Law of Excluded Middle.

The problem with this list is that it is *very* restrictive and it is even redundant. For example, the Law of Identity can mean either 'For all x, x=x' OR it can mean 'for all propositions, p<=>p'. Both are true, but the second is from propositional logic and the first from the logic of equality. But, from these, you cannot even show equality satisfies x=y & y=z => x=z. You can't show that x=y => y=x. Those are additional assumptions about equality that need to be made.

Next, in propositional logic, the term 'p implies q' is logically equivalent to '(not p) or q'. So, the claim that p=>p is logically identical to (not p) or p, in other words, the law of excluded middle! It is a definitional thing. And the Law of Non-Contradiction is simply the statement ' not (p and (not p)). But guess what? (p and q) is *defined* to be not ((not p) or (not q)), so once again, the LNC is exactly the same statement as the LEM (as long as you know that not(not p) =>p, i.e, if it is false that p is false, then p is true---hmmm....that's another logical statement that was left out).

Next, nothing was assumed as a rule of inference. Typically, Modus Tolens is assumed: if we know p and we know p=>q, then we can conclude q.

Nothing was assumed about quantifiers: 'there exists' and 'for every'. Typically, logic assumes that the negation of an existence statement is the same as saying 'for all x, the statement is false'. This is yet another basic logic axiom that is left out of the list.

Finally, with this list, it is impossible to do math at all because no method is given for producing collections. That is yet another collection of assumptions concerning how collections can be constructed. if you want the natural numbers, you have to allow *some* sort of set theory. But that goes *way* beyond the list of 'basic' assumptions made, even for finite sets.

So, I would strongly suggest you take a *modern* logic course that covers propositional and quantifier logic and *then* look at the whole host of axioms needed to construct even finite set theory. Your claim that the axiom of Infinity is suspect is very strange given the other axioms that are required to even *start* talking about addition and multiplication.

And let's face it, division is NOT always defined: 5/0 is not a well defined object. Even ordinary division requires an extension into fractions, which were quite far from being 'self-evident' to many people in history (Even Aristotle). The point is what we might consider to be 'absurd' today is likely not to be so once we get over our biases and realize there are no actual contradictions involved.

Finally, there seems to be a bias against 'abstract' set theory. I would suggest that is misplaced. By showing the abstract ideas have no internal contradictions, we see the range of *logical* possibilities and find that range is much, much broader than those suggesting actual infinites are problematic.