4 added 54 characters in body; deleted 5 characters in body

First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight.

I just learned in Sergey Melikhov's answer to another question something about the Axiom of Determinacy (AD). This Axiom is equivalent to the statement:

$\forall G \subseteq Seq(S):$

$$\forall a \in S :\exists a' \in S :\forall b \in S :\exists b' \in S :\forall c \in S :\exists c' \in S ... : (a,a',b,b',c,c'...) \in G$$ or $$\exists a \in S :\forall a' \in S :\exists b \in S :\forall b' \in S :\exists c \in S :\forall c' \in S ... :(a,a',b,b',c,c'...) \notin G$$

where $Seq(S)$ is the set of all $\omega$-sequences of some countable set $S$. It is thus some infinitary generalization of de Morgan's rule and seems rather natural. $ZF+AD$ seems to be rather realistic (in my opinion) version of set theory, in the sense that it avoids many unphysical paradoxes (such as non-measurable sets, paradoxical decompositions, non-continuous linear functions on Banach spaces etc.) Still, it seems to be strong enough to reproduce enough infinitary mathematics, so that a development a lot of mathematics and of theoretical physics etc. is possible.

It is a fact that $ZF+AD$ can prove consistency of $ZF$. Basically, the question is whether one can continue this process. The question is more precisely:

Question: Is there a hierarchy, which consists of an axiom $AD_{\alpha}$ for each ordinal some ordinals $\alpha$ (inspired maybe by some version of de Morgan's laws for ordinal sequences of quantifiers applied to sets of sequences in sets of some cardinality), such that $AD=AD_{\omega}$ AD=AD_{1}$and$ZF + \cup_{\beta \leq \alpha} ZF_{\beta}$can prove consistency of$ZF + \cup_{\beta < \alpha} ZF_{\beta}$. If that is the case, why not taking$ZF + \cup_{\alpha} AD_{\alpha}$as the axiomatic foundation of mathematics. The bad thing about this would be that the axioms do not form a set, the payoff would be that it proves the consistency of itself and is philosophically sound. EDIT: From Emil Jeřábek's comment I understand that the question does not make sense as stated since there are only countably many formulas and hence there cannot be uncountably many axioms. So the right (and more modest) question to ask would maybe be the following: Question: Does there exist an equally natural axiom$AD'$(based again on some infinitary version of a well-known principle like de Morgan's laws) which proves (together with$ZF+AD$) consistency$ZF + AD$? Maybe there is also some infinitary version of set theory which allows for sentences of arbitrary length like the one which was used above to describe the meaning of$AD$. This could be the place, where the first question could have an answer. 3 added 712 characters in body; added 24 characters in body First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight. I just learned in Sergey Melikhov's answer to another question something about the Axiom of Determinacy (AD). This Axiom is equivalent to the statement:$\forall G \subseteq Seq(S):$$$\forall a \in S :\exists a' \in S :\forall b \in S :\exists b' \in S :\forall c \in S :\exists c' \in S ... : (a,a',b,b',c,c'...) \in G$$ or $$\exists a \in S :\forall a' \in S :\exists b \in S :\forall b' \in S :\exists c \in S :\forall c' \in S ... :(a,a',b,b',c,c'...) \notin G$$ where$Seq(S)$is the set of all$\omega$-sequences of some countable set$S$. It is thus some infinitary generalization of de Morgan's rule and seems rather natural.$ZF+AD$seems to be rather realistic (in my opinion) version of set theory, in the sense that it avoids many unphysical paradoxes (such as non-measurable sets, paradoxical decompositions, non-continuous linear functions on Banach spaces etc.) Still, it seems to be strong enough to reproduce enough infinitary mathematics, so that a development a lot of mathematics and of theoretical physics etc. is possible. It is a fact that$ZF+AD$can prove consistency of$ZF$. Basically, the question is whether one can continue this process. The question is more precisely: Question: Is there a hierarchy, which consists of an axiom$AD_{\alpha}$for each ordinal$\alpha$(inspired maybe by some version of de Morgan's laws for ordinal sequences of quantifiers applied to sets of sequences in sets of some cardinality), such that$AD=AD_{\omega}$and$ZF + \cup_{\beta \leq \alpha} ZF_{\beta}$can prove consistency of$ZF + \cup_{\beta < \alpha} ZF_{\beta}$. If that is the case, why not taking$ZF + \cup_{\alpha} AD_{\alpha}$as the axiomatic foundation of mathematics. The bad thing about this would be that the axioms do not form a set, the payoff would be that it proves the consistency of itself and is philosophically sound. EDIT: From Emil Jeřábek's comment I understand that the question does not make sense as stated since there are only countably many formulas and hence there cannot be uncountably many axioms. So the right (and more modest) question to ask would maybe be the following: Question: Does there exist an equally natural axiom$AD'$(based again on some infinitary version of a well-known principle like de Morgan's laws) which proves (together with$ZF+AD$) consistency$ZF + AD$? Maybe there is also some infinitary version of set theory which allows for sentences of arbitrary length like the one which was used above to describe the meaning of$AD$. This could be the place, where the first question could have an answer. 2 edited body First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight. I just learned in Sergey Melikhov's answer to another question something about the Axiom of Determinacy (AD). This Axiom is equivalent to the statement:$\forall G \subseteq Seq(S):$$$\forall a \in S :\exists a' \in S :\forall b \in S :\exists b' \in S :\forall c \in S :\exists c' \in S ... : (a,a',b,b',c,c'...) \in G$$ or $$\exists a \in S :\forall a' \in S :\exists b \in S :\forall b' \in S :\exists c \in S :\forall c' \in S ... :(a,a',b,b',c,c'...) \notin G$$ where$Seq(S)$is the set of all$\omega$-sequences of some countable set$S$. It is thus some infinitary generalization of de Morgan's rule and seems rather natural.$ZF+AD$seems to be rather realistic (in my opinion) version of set theory, in the sense that it avoids many unphysical paradoxes (such as non-measurable sets, paradoxical decompositions, non-continuous linear functions on Banach spaces etc.) Still, it seems to be strong enough to reproduce enough infinitary mathematics, so that a development a lot of mathematics and of theoretical physics etc. is possible. It is a fact that$ZF+AD$can prove consistency of$ZF$. Basically, the question is whether one can continue this process. The question is more precisely: Question: Is there a hierarchy, which consists of an axiom$AD_{\alpha}$for each ordinal$\alpha$(inspired maybe by some version of de Morgan's laws for ordinal sequences of quantifiers applied to sets of sequences in sets of some cardinality), such that$AD=AD_{\omega}$and$ZF + \cup_{\beta \leq \alpha} ZF_{\beta}$can prove consistency of$ZF + \cup_{\beta < \alpha} ZF_{\beta}$. If that is the case, why not taking$ZF + \cup_{\alpha} AD_{\alpha}\$ as the axiomatic foundation of mathematics. The bad thing about this would be that the axioms do not form a set, the payoff would be that it proves its the consistency of itself and is philosophically sound.

1