# Any paradoxical theorems arising from large cardinal axioms?

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.

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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.

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(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.

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Andres, welcome to MO! – Joel David Hamkins May 16 '10 at 2:11
Andres, I admit that today's set theorists understand large cardinals much better than their predecessors. I just want to highlight how much our understanding has changed (and therefore, how much it could change in the future). – John Stillwell May 16 '10 at 2:52
Hi Joel! Oh, John, I agree. Determinacy itself is a good example of the same phenomenon. From being a somewhat ad hoc theory about infinite games, it has gone to become a systematic way of studying and establishing regularity properties. – Andrés E. Caicedo May 16 '10 at 14:30
I believe that, back when supercompact cardinals were a new invention, Solovay mentioned in print that they might imply L[R]-determinacy (which turned out to be true, even with weaker large cardinals). Nevertheless, I believe Andres's description of the situation later, in the early 80's, is accurate. Determinacy had, in the meantime, started to look much stronger. – Andreas Blass Jul 3 '10 at 20:36

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.

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