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The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an interesting result of Hamkins and Gitman, Hamkins, Johnstone if we delete the axiom of power set from $ZFC$ then it is possible that $\omega_{1}$ be a singular cardinal. Now a natural question is:

Question: For which one of the axioms of $ZFC$ like $A$ we have:

$Con(ZFC)\Longrightarrow Con(ZFC-A+cf(\omega_{1})<\omega_{1})$

Particularly is the following statement true?

$Con(ZFC)\Longrightarrow Con(ZFC-Rep+cf(\omega_{1})<\omega_{1})$

Note that in some sense the axioms of power set and replacement are the most "powerful" axioms of $ZFC$. Also they have a "dual" role in mathematics.

The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an interesting result of Hamkins and Gitman if we delete the axiom of power set from $ZFC$ then it is possible that $\omega_{1}$ be a singular cardinal. Now a natural question is:

Question: For which one of the axioms of $ZFC$ like $A$ we have:

$Con(ZFC)\Longrightarrow Con(ZFC-A+cf(\omega_{1})<\omega_{1})$

Particularly is the following statement true?

$Con(ZFC)\Longrightarrow Con(ZFC-Rep+cf(\omega_{1})<\omega_{1})$

Note that in some sense the axioms of power set and replacement are the most "powerful" axioms of $ZFC$. Also they have a "dual" role in mathematics.

The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an interesting result of Gitman, Hamkins, Johnstone if we delete the axiom of power set from $ZFC$ then it is possible that $\omega_{1}$ be a singular cardinal. Now a natural question is:

Question: For which one of the axioms of $ZFC$ like $A$ we have:

$Con(ZFC)\Longrightarrow Con(ZFC-A+cf(\omega_{1})<\omega_{1})$

Particularly is the following statement true?

$Con(ZFC)\Longrightarrow Con(ZFC-Rep+cf(\omega_{1})<\omega_{1})$

Note that in some sense the axioms of power set and replacement are the most "powerful" axioms of $ZFC$. Also they have a "dual" role in mathematics.

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user36136
user36136

The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an interesting result of Hamkins and Gitman if onewe delete the axiom of power set from $ZFC$ then it is possible that $\omega_{1}$ be a singular cardinal. Now a natural question is:

Question: For which one of the axioms of $ZFC$ like $A$ we have:

$Con(ZFC)\Longrightarrow Con(ZFC-A+cf(\omega_{1})<\omega_{1})$

Particularly is the following statement true?

$Con(ZFC)\Longrightarrow Con(ZFC-Rep+cf(\omega_{1})<\omega_{1})$

Note that in some sense the axioms of power set and replacement are the most "powerful" axioms of $ZFC$. Also they have a "dual" role in mathematics.

The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an interesting result of Hamkins and Gitman if one delete the axiom of power set from $ZFC$ then it is possible that $\omega_{1}$ be a singular cardinal. Now a natural question is:

Question: For which one of the axioms of $ZFC$ like $A$ we have:

$Con(ZFC)\Longrightarrow Con(ZFC-A+cf(\omega_{1})<\omega_{1})$

Particularly is the following statement true?

$Con(ZFC)\Longrightarrow Con(ZFC-Rep+cf(\omega_{1})<\omega_{1})$

Note that in some sense the axioms of power set and replacement are the most "powerful" axioms of $ZFC$. Also they have a "dual" role in mathematics.

The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an interesting result of Hamkins and Gitman if we delete the axiom of power set from $ZFC$ then it is possible that $\omega_{1}$ be a singular cardinal. Now a natural question is:

Question: For which one of the axioms of $ZFC$ like $A$ we have:

$Con(ZFC)\Longrightarrow Con(ZFC-A+cf(\omega_{1})<\omega_{1})$

Particularly is the following statement true?

$Con(ZFC)\Longrightarrow Con(ZFC-Rep+cf(\omega_{1})<\omega_{1})$

Note that in some sense the axioms of power set and replacement are the most "powerful" axioms of $ZFC$. Also they have a "dual" role in mathematics.

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user36136
user36136

On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$

The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an interesting result of Hamkins and Gitman if one delete the axiom of power set from $ZFC$ then it is possible that $\omega_{1}$ be a singular cardinal. Now a natural question is:

Question: For which one of the axioms of $ZFC$ like $A$ we have:

$Con(ZFC)\Longrightarrow Con(ZFC-A+cf(\omega_{1})<\omega_{1})$

Particularly is the following statement true?

$Con(ZFC)\Longrightarrow Con(ZFC-Rep+cf(\omega_{1})<\omega_{1})$

Note that in some sense the axioms of power set and replacement are the most "powerful" axioms of $ZFC$. Also they have a "dual" role in mathematics.