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Asaf Karagila ♦
Asaf Karagila ♦
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I Are there any examples of two large cardinal axioms AX$AX$ and AY$AY$, in the language of first order ZFC$ZFC$, which which satisfy the following conditions. (1) Each of them defines a unique cardinal number-C(AX) for AX and C(AY) for AY-not like the axiom of measurable cardinals wich defines a whole collection of cardinal numbers. (2) If T denotes the theory ZFC+AX+AY, then T has not yet been proved inconsistent. (3) T proves that each of C(AX) and C(AY) is larger that the smallest strongly inacessible cardinal number. (4) The sentences of T stating that C(AX)<C(AY) and that C(AY)<C(AX) are each consistent with T, if T is consistent.

  1. Each of them defines a unique cardinal number - $C(AX)$ for $AX$ and $C(AY)$ for $AY$ - not like the axiom of measurable cardinals which defines a whole collection of cardinal numbers.

  2. If $T$ denotes the theory $ZFC+AX+AY$, then $T$ has not yet been proved inconsistent.

  3. $T$ proves that each of $C(AX)$ and $C(AY)$ is larger that the smallest strongly inaccessible cardinal number.

  4. The sentences of $T$ stating that $C(AX) < C(AY)$ and that $C(AY) < C(AX)$ are each consistent with $T$, if $T$ is consistent.

I Are there any examples of two large cardinal axioms AX and AY, in the language of first order ZFC, which satisfy the following conditions. (1) Each of them defines a unique cardinal number-C(AX) for AX and C(AY) for AY-not like the axiom of measurable cardinals wich defines a whole collection of cardinal numbers. (2) If T denotes the theory ZFC+AX+AY, then T has not yet been proved inconsistent. (3) T proves that each of C(AX) and C(AY) is larger that the smallest strongly inacessible cardinal number. (4) The sentences of T stating that C(AX)<C(AY) and that C(AY)<C(AX) are each consistent with T, if T is consistent.

Are there any examples of two large cardinal axioms $AX$ and $AY$, in the language of first order $ZFC$, which satisfy the following conditions.

  1. Each of them defines a unique cardinal number - $C(AX)$ for $AX$ and $C(AY)$ for $AY$ - not like the axiom of measurable cardinals which defines a whole collection of cardinal numbers.

  2. If $T$ denotes the theory $ZFC+AX+AY$, then $T$ has not yet been proved inconsistent.

  3. $T$ proves that each of $C(AX)$ and $C(AY)$ is larger that the smallest strongly inaccessible cardinal number.

  4. The sentences of $T$ stating that $C(AX) < C(AY)$ and that $C(AY) < C(AX)$ are each consistent with $T$, if $T$ is consistent.

I typed in condition (4) which had got deleted.
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Garabed Gulbenkian
Garabed Gulbenkian
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I Are there any examples of two large cardinal axioms AX and AY, in the language of first order ZFC, which satisfy the following conditions. (1) Each of them defines a unique cardinal number-C(AX) for AX and C(AY) for AY-not like the axiom of measurable cardinals wich defines a whole collection of cardinal numbers. (2) If T denotes the theory ZFC+AX+AY, then T has not yet been proved inconsistent. (3) T proves that each of C(AX) and C(AY) is larger that the smallest strongly inacessible cardinal number. (4) The sentences of T stating that C(AX)<C(AY) and that C(AY)<C(AX) are each consistent with T, if T is consistent.

Are there any examples of two large cardinal axioms AX and AY, in the language of first order ZFC, which satisfy the following conditions. (1) Each of them defines a unique cardinal number-C(AX) for AX and C(AY) for AY-not like the axiom of measurable cardinals wich defines a whole collection of cardinal numbers. (2) If T denotes the theory ZFC+AX+AY, then T has not yet been proved inconsistent. (3) T proves that each of C(AX) and C(AY) is larger that the smallest strongly inacessible cardinal number. (4) The sentences of T stating that C(AX)<C(AY) and that C(AY)<C(AX) are each consistent with T, if T is consistent.

I Are there any examples of two large cardinal axioms AX and AY, in the language of first order ZFC, which satisfy the following conditions. (1) Each of them defines a unique cardinal number-C(AX) for AX and C(AY) for AY-not like the axiom of measurable cardinals wich defines a whole collection of cardinal numbers. (2) If T denotes the theory ZFC+AX+AY, then T has not yet been proved inconsistent. (3) T proves that each of C(AX) and C(AY) is larger that the smallest strongly inacessible cardinal number. (4) The sentences of T stating that C(AX)<C(AY) and that C(AY)<C(AX) are each consistent with T, if T is consistent.

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Garabed Gulbenkian
Garabed Gulbenkian
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A question about the comparability of large cardinals.

Are there any examples of two large cardinal axioms AX and AY, in the language of first order ZFC, which satisfy the following conditions. (1) Each of them defines a unique cardinal number-C(AX) for AX and C(AY) for AY-not like the axiom of measurable cardinals wich defines a whole collection of cardinal numbers. (2) If T denotes the theory ZFC+AX+AY, then T has not yet been proved inconsistent. (3) T proves that each of C(AX) and C(AY) is larger that the smallest strongly inacessible cardinal number. (4) The sentences of T stating that C(AX)<C(AY) and that C(AY)<C(AX) are each consistent with T, if T is consistent.