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2 corrected references and other minor improvements

As I pointed out in the meta thread, this question overlaps with a bunch of older MO questions.

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 section III §III of Believing the axiomsAxioms, I [JSL 53 (1988), 736-764481-511, MR0947855]. 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. 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 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]). 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]. Of course, detailed information can be found in Kanamori's The Higher Infinite [Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994].

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As I pointed out in the meta thread, this question overlaps with a bunch of older MO questions.

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 section III of Believing the axioms, I [JSL 53 (1988), 736-764, MR0947855]. In this wonderful paper, Maddy justifies 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.

• The maximization argument. The maximize rule of thumb is perhaps best understood as the opposite of Occam's 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 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 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]). 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].