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In the same spirit of this question:

How much of mathematical General Relativity depends on the Axiom of Choice?

I want to go radically further ahead and ask for what remains of mathematical general relativity in case one limits one's set theory to that of ZF or at most ZF+ADC(axiom of dependent choice).

The motivation is that: to cause any departure from common sense, such as Banach-Tarski Paradox the minimum known set theory needed is ZF+HB for the moment, and at the same time one knows that BT cannot be deduced from ZF+ADC.

So simply I want to know which results depend crucially on the Hahn-Banach theorem(axiom) part of the ZF+HB, i.e. the results that exist in ZF+HB, but fail to exist in ZF or at most ZF+ADC.

PS: ZF<ZF+ADC<ZF+HB<ZF+UFL<ZF+AC

Where

UFL:Ultra Filter lemma

AC:Axiom of choice

ADC:Axiom of dependent choice

HB:Hahn-Banach

In the same spirit of this question:

How much of mathematical General Relativity depends on the Axiom of Choice?

I want to go radically further ahead and ask for what remains of mathematical general relativity in case one limits one's set theory to that of ZF.

The motivation is that: to cause any departure from common sense, such as Banach-Tarski Paradox the minimum known set theory needed is ZF+HB for the moment, and at the same time one knows that BT cannot be deduced from ZF.

So simply I want to know which results depend crucially on the Hahn-Banach theorem(axiom) part of the ZF+HB, i.e. the results that exist in ZF+HB, but fail to exist in ZF.

PS: ZF<ZF+HB<ZF+UFL<ZF+AC

Where

UFL:Ultra Filter lemma

AC:Axiom of choice

HB:Hahn-Banach

In the same spirit of this question:

How much of mathematical General Relativity depends on the Axiom of Choice?

I want to go radically further ahead and ask for what remains of mathematical general relativity in case one limits one's self set theory to that of ZF or at most ZF+ADC(axiom of dependent choice).

The motivation is that: to cause any departure from common sense, such as Banach-Tarski Paradox the minimum known set theory needed is ZF+HB for the moment, and at the same time one knows that BT cannot be deduced from ZF+ADC.

So simply I want to know which results depend crucially on the Hahn-Banach theorem(axiom) part of the ZF+HB, i.e. the results that exist in ZF+HB, but fail to exist in ZF+ADC.

PS: ZF+ADC<ZF+HB<ZF+UFL<ZF+AC

Where

UFL:Ultra Filter lemma

AC:Axiom of choice

ADC:Axiom of dependent choice

HB:Hahn-Banach

In the same spirit of this question:

How much of mathematical General Relativity depends on the Axiom of Choice?

I want to go radically further ahead and ask for what remains of mathematical general relativity in case one limits one's set theory to that of ZF or at most ZF+ADC(axiom of dependent choice).

The motivation is that: to cause any departure from common sense, such as Banach-Tarski Paradox the minimum known set theory needed is ZF+HB for the moment, and at the same time one knows that BT cannot be deduced from ZF+ADC.

So simply I want to know which results depend crucially on the Hahn-Banach theorem(axiom) part of the ZF+HB, i.e. the results that exist in ZF+HB, but fail to exist in ZF or at most ZF+ADC.

PS: ZF<ZF+ADC<ZF+HB<ZF+UFL<ZF+AC

Where

UFL:Ultra Filter lemma

AC:Axiom of choice

ADC:Axiom of dependent choice

HB:Hahn-Banach

In the same spirit of this question:

How much of mathematical General Relativity depends on the Axiom of Choice?

I want to go radically further ahead and ask for what remains of mathematical general relativity in case one limits one's self set theory to that of ZF or at most ZF+ADC(axiom of dependent choice).

The motivation is that: to cause any departure from common sense, such as Banach-Tarski Paradox the minimum known set theory needed is ZF+HB for the moment, and at the same time one knows that BT cannot be deduced from ZF+ADC.

So simply I want to know which results depend crucially on the Hahn-Banach theorem(axiom) part of the ZF+HB. The results that exit in ZF+HB but fail to exists in ZF+ADC.

PS: ZF+ADC<ZF+HB<ZF+UFL<ZF+AC

Where

UFL:Ultra Filter lemma

AC:Axiom of choice

ADC:Axiom of dependent choice

HB:Hahn-Banach

In the same spirit of this question:

How much of mathematical General Relativity depends on the Axiom of Choice?

I want to go radically further ahead and ask for what remains of mathematical general relativity in case one limits one's self set theory to that of ZF or at most ZF+ADC(axiom of dependent choice).

The motivation is that: to cause any departure from common sense, such as Banach-Tarski Paradox the minimum known set theory needed is ZF+HB for the moment, and at the same time one knows that BT cannot be deduced from ZF+ADC.

So simply I want to know which results depend crucially on the Hahn-Banach theorem(axiom) part of the ZF+HB, i.e. the results that exist in ZF+HB, but fail to exist in ZF+ADC.

PS: ZF+ADC<ZF+HB<ZF+UFL<ZF+AC

Where

UFL:Ultra Filter lemma

AC:Axiom of choice

ADC:Axiom of dependent choice

HB:Hahn-Banach