Actual Infinity in Reality?

[polymath257]

I'd point out that *finite* sets in mathematics are just as 'abstract'. mathematics is the study of formal axiom systems. So of course it is all abstract.

1. Yes, but finite sets are countable and we can know every member. An infinite set is not countable, unbounded. It is more abstract since we cannot find examples in reality to stand in as a comparison. Also, finite sets are defined by logical axioms which are more self-evident, where infinite sets are defined by non-logical axioms, which is interchangeable with 'assumption'. 

The idea that axioms are 'intuitively obvious' is another outdated idea. Euclid tried that with Euclidean geometry, but we have found that his system, even of geometry, was far from unique. Non-euclidean geometry is equally consistent, but gives different answers.

Which leads to the point: when there are different axioms systems that are all consistent, there is nothing to say which is correct and which is not correct except to go to observation and testing.

So, the fact that introducing actual infinities does bring a contradiction shows that there is no *logical* reason to exclude actual infinities.

2. So you cannot mathematically prove an actual infinity. All you have in mathematics is an axiom that assumes an actual infinity. Mathematics cannot say one way or another whether an actual infinity exists. Therefore, you must move to observations/tests with real objects and see if the concept can stand up to scrutiny. There are a whole series of absurdities that you run into when you start to do thought experiments with infinities. So while mathematics does not show a logical contradiction, observation does. 

And I agree--the actual existence of actual infinites has not been proven. But that isn't my claim. My claim is that there is no *logical* issue with them and that they should be considered as one *possibility*. And that is quite enough to destroy the Kalam argument. There is no *contradiction* with having an infinite regress of causes. It is internally consistent and so cannot be dismissed out of hand.

3. No, your claim is that there is no *mathematical* issue. You cannot say there is no *logical* issue when all we have are paradoxes and absurdities when we think about an infinite number of real objects. Mathematics is definitely not the only source of logic.

1. There is a technical terminology in mathematics: an infinite set is called countable if it can be put into a one-to-one correspondence with the set of natural numbers. Your usage of the word 'countable' is different than the mathematical usage, which is now standard usage (for a reason, I might add).

An infinite set is simply one that is not finite. In the terminology that is now standard, infinite sets can also be countable or uncountable.

Your claim that finite sets are defined through 'logical axioms' in a way that is substantially different than the assumptions for infinite sets is unfounded. Mathematicians have gotten away from the naive idea of axioms as being 'self-evident truths'. Why? Because 'self-evident' is ill defined and varies from person to person. Instead, mathematics deals with *internally consistent sets of axioms*. Those for finite sets are not more logical than those for infinite sets. They are simply different logical systems.

Technically, the term 'logic' only applies to propositional and quantifier logic: words like and, or, not, implies, for every, there exist. ANYTHING past that goes beyond logic. And yes, that means even finite sets have non-logical axioms because they have non-logical concepts (i.e, those going beyond the previous ones, like set membership). That doesn't mean the new concepts are *against* logic. They are *extensions* of the basic logic of propositions and quantifiers.

But the systems built up still have to be free of internal contradictions. And yes, those systems that have infinite sets *are* free of such contradictions.

2. Yes, the assumption of an infinite set is an additional axiom. But so are others like the Axiom of Choice. The freedom from internal contradictions shows the ideas to be coherent and NOT logical impossibilities.

I would suggest the 'absurdities' that arise when you *first* start thinking about infinities are the result of not understanding that infinite sets and finite sets have different properties. But that is to be expected.

For example, exactly what in the Hilbert Hotel do you consider to be an 'absurdity'? That if it is full, the people in the rooms can be rearranged to accommodate another guest? How, precisely, is that an absurdity? If anything, it serves to show how, if there was an infinite amount of time prior to the present, there is also an infinite amount of time prior to a minute from now. Nothing *physical* is violated in this 'absurdity'.

3. The fact that there are no *logical* contradictions, as shown by the consistency of the mathematics, is quite enough to show there are no 'logical' problems with an actual infinity. No, math isn't the only source of logic (if anything, logic is a sub-theory of math), but the internal consistency of the ideas shows it to be without logical issues.

And, once again, you have failed to show an *observational* issue with an actual infinity. Space may well be infinite in extent. There is no observational evidence otherwise. it is quite possible that in such an infinite space, there are an infinite number of objects. Once again, there is absolutely no observational evidence at this point to say otherwise. If anything, the observational evidence points to such an actual infinity of space (from the observation of nearly flat space).

Once again, I ask you to point to the *actual absurdities* of an actual infinity, say either of time prior to ours or of space. What *physical* facts are violated by either?