Revisions to Why should we believe in the axiom of regularity?

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edited Jun 25 at 18:32

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Today I started reading Maddy's Believing the axioms Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity.

Apparently, the reason why we usually take it is because it makes sets well-founded and makes $\in$-induction work, or because it puts all sets into a hierarchy (namely $V$). However, these reasons sound to me more like "we take this, because it's convenient". Another reason commonly given is "It's difficult to think of a set which is an element of itself". This is not a good reason, because many things are difficult to think of, and also one could argue that a set represented by $\{\{\{...\}\}\}$ should do the trick.

That brings me to my question:

Are there any "philosophical" reasons to believe that the axiom of regularity holds?

I understand that this question is quite vague and maybe too broad, but I will be thankful for any responses. ${}{}$

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity.

Apparently, the reason why we usually take it is because it makes sets well-founded and makes $\in$-induction work, or because it puts all sets into a hierarchy (namely $V$). However, these reasons sound to me more like "we take this, because it's convenient". Another reason commonly given is "It's difficult to think of a set which is an element of itself". This is not a good reason, because many things are difficult to think of, and also one could argue that a set represented by $\{\{\{...\}\}\}$ should do the trick.

That brings me to my question:

Are there any "philosophical" reasons to believe that the axiom of regularity holds?

I understand that this question is quite vague and maybe too broad, but I will be thankful for any responses. ${}{}$

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity.

Apparently, the reason why we usually take it is because it makes sets well-founded and makes $\in$-induction work, or because it puts all sets into a hierarchy (namely $V$). However, these reasons sound to me more like "we take this, because it's convenient". Another reason commonly given is "It's difficult to think of a set which is an element of itself". This is not a good reason, because many things are difficult to think of, and also one could argue that a set represented by $\{\{\{...\}\}\}$ should do the trick.

That brings me to my question:

Are there any "philosophical" reasons to believe that the axiom of regularity holds?

I understand that this question is quite vague and maybe too broad, but I will be thankful for any responses. ${}{}$

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edited Jun 25 at 18:17

Martin Sleziak

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edited Apr 9, 2019 at 8:13

Denis Serre

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Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity.

Apparently, the reason why we usually take it is because it makes sets well-founded and makes $\in$-induction work, or because it puts all sets into a hierarchy (namely $V$). However, these reasons sound to me more like "we take this, because it's convenient". Another reason commonly given is "It's difficult to think of a set which is an element of itself". This is not a good reason, because many things are difficult to think of, and also one could argue that a set represented by $\{\{\{...\}\}\}$ should do the trick.

That brings me to my question:

Are there any "philosophical" reasons to believe that the axiom of regularity holds?

I understand that this question is quite vague and maybe too broad, but I will be thankful for any responses. ${}{}$

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity.

Apparently, the reason why we usually take is because it makes sets well-founded and makes $\in$-induction work, or because it puts all sets into a hierarchy (namely $V$). However, these reasons sound to me more like "we take this, because it's convenient". Another reason commonly given is "It's difficult think of a set which is an element of itself". This is not a good reason, because many things are difficult to think of, and also one could argue that a set represented by $\{\{\{...\}\}\}$ should do the trick.

That brings me to my question:

Are there any "philosophical" reasons to believe that the axiom of regularity holds?

I understand that this question is quite vague and maybe too broad, but I will be thankful for any responses. ${}{}$

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity.

Apparently, the reason why we usually take it is because it makes sets well-founded and makes $\in$-induction work, or because it puts all sets into a hierarchy (namely $V$). However, these reasons sound to me more like "we take this, because it's convenient". Another reason commonly given is "It's difficult to think of a set which is an element of itself". This is not a good reason, because many things are difficult to think of, and also one could argue that a set represented by $\{\{\{...\}\}\}$ should do the trick.

That brings me to my question: