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I'm curious to know the history of the interaction between large cardinals and traveling to (creating) universes through forcing. The question arose because I understand that Peano Arithmatists force, but it is like a large cardinal embedding in that it forces the existance of an inner model. When was the first large cardinal introduced and was it viewed as a way to create a model of set theory? Were large cardinals considered before Cohen first forced?

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Is this related to the recent pope election? –  copper.hat Mar 16 '13 at 1:04
Hehe...so I'm not the only one! –  DonAntonio Mar 16 '13 at 1:05
Maybe if I had asked if infallible cardinals exist after 2013:) I don't get the joke about the continuum hypothesis. –  Erin K Carmody Mar 16 '13 at 1:42
The joke has been erased. –  Erin K Carmody Mar 17 '13 at 20:38
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1 Answer

up vote 5 down vote accepted

Very much so.

Measurable cardinals were considered by Banach and Kuratowski in 1929 and 1930. In 1943 Tarski proved that measurable cardinals are weakly compact, although the definition by tree properties and by coloring were given only in 1961 by Erdos and Tarski.

Hausdorff worked with weakly inaccessible cardinals, and Mahlo defined what is now known as Mahlo cardinals, Sierpinski and Tarski defined strongly inaccessible cardinals, as well Zermelo worked with them (e.g. he proved that the only models of $\sf ZFC_2$ are $V_\kappa$ for strongly inaccessible $\kappa$). All these things were well before the 1950's.

All this is just from the brief historical remarks in Jech's Set Theory (Chapters 9 and 10), brief excerpts from the introduction of Kanamori's wonderful The Higher Infinite.

Scott's theorem, by the way, that a measurable cardinal implies $V\neq L$ was published in 1961.

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However, were they known at the time, or at least suspected, to be large cardinals in the sense of being cardinals whose existence is not provable? –  Zhen Lin Mar 16 '13 at 1:08
@Zhen Lin: That is a good question, I'll read further. But it is a different question than the one I read above... –  Asaf Karagila Mar 16 '13 at 1:11
Zhen, I believe so. If Zermelo proved that $V_\kappa$ is a model of $\sf ZFC_2$, then it is certainly a model of $\sf ZFC$. By the early 1930's it was already known that $\sf ZFC$ cannot prove its own consistency. –  Asaf Karagila Mar 16 '13 at 1:13
Thanks! I really wanted to know the story, and looked here for a quick answer, very helpful. –  Erin K Carmody Mar 16 '13 at 1:39
@ZhenLin we can't even say that large cardinals ever existed, just that if we (consistent ZFC models) exist, then worldly, incaccessible, Mahlo, measurable, and so on exist and then we can see how the worlds play. I'm glad you pointed this out. –  Erin K Carmody Mar 17 '13 at 20:37
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