(February 26, 2018 at 4:24 pm)polymath257 Wrote: 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.
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'.
Quote:The idea that axioms are 'intuitively obvious' is another outdated idea.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.
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.
Quote: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.
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.