[Here](http://logiciansdoitwithmodels.com/2011/08/17/fom-posting-on-a-kiselevs-claim-that-there-are-no-weakly-inaccessible-cardinals-in-zf/) it is mentioned that someone claims to have proven that there are no weakly inaccessibles in ZF.


**Question 1:** What reasons are there to believe that weakly inaccessibles exist?


**Question(s) 2:** Since all large cardinals are weakly inaccessible, this would have a profound effect on set theory. What are some of the most significant results *whose only known proof* assumes the existence of weakly inaccessibles? Might any of the arguments go through without their existence? For example, I've heard that the original proof of Fermat's Last Theorem (FLT) assumed (something equivalen to) a large cardinal, but it was then shown that the argument went through without such an assumption.


**Edit.** I just added the phrase "whose only known proof" to Question 2 above, which is what I intended originally. The point of course, is that I want to know which results, if any, would be "lost" if weakly inaccessibles were lost. FLT is not an example of that, but would have been before it was known that weakly inaccessibles are not necessary in its proof.



Recent claim that inaccessibles are inconsistent with ZF
As I pointed out in the meta thread, this question overlaps with a bunch of older MO questions.


* [Arguments against large cardinals](https://mathoverflow.net/questions/44095/arguments-against-large-cardinals)
* [Nonessential use of large cardinals](https://mathoverflow.net/questions/48851/nonessential-use-of-large-cardinals)
* [Inaccessible cardinals and Andrew Wiles's proof](https://mathoverflow.net/questions/35746/inaccessible-cardinals-and-andrew-wiless-proof)
* [Reasons to believe Vopenka's principle/huge cardinals are consistent](https://mathoverflow.net/questions/29302/reasons-to-believe-vopenkas-principle-huge-cardinals-are-consistent)


However, none of these questions directly address the particular case of the existence of inaccessible cardinals, which is of special interest as this is the weakest of all large cardinal hypotheses. This answer focuses on that case.


Penelope Maddy gives several answers to Question 1 in §III of *Believing the Axioms, I* [JSL 53 (1988), 481-511, [MR0947855](http://www.ams.org/mathscinet-getitem?mr=947855)].
In this wonderful paper, Maddy justifies many set theoretic axioms and hypotheses using five widely believed "rules of thumb": *maximize*, *inexhaustibility*, *uniformity*, *whimsical identity*, and *reflection*. Here is a brief summary of these five arguments as it pertains to the existence of inaccessible cardinals.


* *The maximization argument.*
The *maximize* rule of thumb is perhaps best understood as the opposite of [Occam's Razor](http://en.wikipedia.org/wiki/Occam%39s_razor).
However, blind application of this easily leads to contradictions.
Thus, the rule is generally understood as a pair of statements: 
*thikness* — powersets are very large; and 
*tallness* — there are lots and lots of ordinals.
The second easily leads to the existence of inaccessibles.
* *The inexhaustibility argument.* 
Maddy describes this one very well: "The universe of sets is too complex to be exhausted by any handful of operations, in particular by power set and replacement, the two given by the axioms of Zermelo and Fraenkel.
Thus there must be an ordinal number after all the ordinals generated by replacement and power set.
This is an inaccessible." (p. 502)
* *The uniformity argument.*
Uniformity basically states that the richness of the universe should not concentrate in a small region, that if a certain property is found at a certain level of the cumulative hierarchy then analogue properties should also be found higher up.
Thus, there should be many cardinals that share the same properties as $\aleph\_0$, such as the fact that $2^k < \aleph\_0$ for every $k < \aleph\_0$.
Combined with regularity, this leads to the existence of inaccessibles.
* *The whimsical identity argument.*
This rule of thumb states that there should be no accidental identities, "like the identity between 'human' and 'featherless biped'." (p. 499)
It seems unlikely that $\aleph\_0$ should be characterized as *the unique regular cardinal $\kappa$ such that $2^\mu < \kappa$ for every $\mu < \kappa$*.
Therefore, there must be inaccessible cardinals.
* *The reflection argument.*
This powerful rule of thumb is a generalization of Montague's Reflection Theorem, which states that for every first-order formula $\phi(\bar{x})$ of $V \vDash \phi(\bar{x})$ then there are arbitrarily large ordinals $\alpha$ such that $V\_\alpha \vDash \phi(\bar{x})$.
The [Reflection Principle](http://en.wikipedia.org/wiki/Reflection_principle) generalizes this from first-order properties to arbitrary properties.
Thus, since $V$ is closed under replacement and powerset, there must be arbitrarily large ordinals $\alpha$ such that $V\_\alpha$ is also closed under replacement and powerset.
These ordinals are inaccessibles.


These five arguments have a lot in common, but the basic principles behind them are quite different.
I would contend that these are five distinct justifications for the existence of inaccessibles.


Note that Maddy's paper has a sequel *Believing the Axioms, II* [JSL 53 (1988), 736-764, [MR0960996](http://www.ams.org/mathscinet-getitem?mr=960996)]).
Let me also point out nother highly relevant paper: Kanamori and Magidor, *The evolution of large cardinal axioms in set theory* [LNM 669, 99-275, [MR0520190](http://www.ams.org/mathscinet-getitem?mr=520190)]. Of course, detailed information can be found in Kanamori's *The Higher Infinite* [Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994].


