Any paradoxical theorems arising from large cardinal axioms? - MathOverflow

Any paradoxical theorems arising from large cardinal axioms?

2010-05-15T23:17:20Z

If we accept the axiom of choice, we take the responsibility of living in a world in which, e.g., a ball in Euclidean 3-space is equiscindable to two isometric copies of itself (Banach-Tarski). So we have a very natural and seemingly harmless axiom (that most matehmaticians are probably willing to accept and use) on one hand, and a quite paradoxical and counterintuitive phenomenon arising from accepting that axiom on the other.

I would like to know if the assumpion of some large cardinal axiom is known to produce some sort of paradoxical phenomena in the "everyday world", as it happens in the case of the axiom of choice.

Answer by Simon Thomas for Any paradoxical theorems arising from large cardinal axioms?

2010-05-15T23:23:37Z

In fact, the opposite is true. If suitable large cardinals exist, then all projective sets of reals (i.e. the definable sets that you are likely to come across in real life) are non-pathological. More precisely, they are measurable, have the Baire property, are either countable or have a perfect subset, etc.

Answer by John Stillwell for Any paradoxical theorems arising from large cardinal axioms?

2010-05-15T23:42:44Z

I broadly agree that large cardinals don't have paradoxical consequences, especially at the "everyday world" level. However, large cardinals have consequences that were initially surprising to set theorists. I think it was a surprise when Scott proved ( in 1961) that measurable cardinals imply the existence of non-constructible sets, and Gödel was astounded that the first measurable cardinal has to be larger that the first inaccessible.

Answer by Andres Caicedo for Any paradoxical theorems arising from large cardinal axioms?

2010-05-16T02:01:16Z

(This was a reply to John that grew beyond the alloted limit.)

large cardinals have consequences that were initially surprising to set theorists.

But, John, this is simply because our intuitions about large cardinals took a bit to form, just as one would expect with any relatively recent theory. Now we understand significantly better how far from $L$ the universe ought to be in the presence of large cardinals, and find the puzzlement at Scott's result curious. We know, for example, that much much much less than a measurable already contradicts $V=L$.

Similarly, when our understanding of large cardinals was relatively poor (early 80s), it was the common belief that for any large cardinal there was a nice inner model with a $\Delta^1_3$ well-ordering of ${\mathbb R}$, and it was by refuting this that we eventually arrived at Woodin cardinals and our current view of the set theoretic universe, where (as Simon mentioned) large cardinals make such well-orderings impossible, since all projective sets are measurable.

(This significant shift in understanding started with the Martin's Maximum paper by Foreman-Magidor-Shelah, Annals of Mathematics, 127 (1988), 1-47, and is very nicely explained in 'Iteration trees' by Martin-Steel, Journal of the American Mathematical Society, 7(1):1–73, 1994.)

This also led, by the way, to the proof that large cardinals imply determinacy in nice inner models, again shifting the prior poorly developed intuitions that had us expect much larger consistency strength for determinacy than it turned out to be the case.