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What axiomatic set theories say that large cardinals exist [closed]
Clash Royale CLAN TAG#URR8PPP
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The title is my question.
I'm curious since I can't seem to find any axiomatic set theory that say that large cardinals exist.
Another thing I’d like to know is that if there are any axiomatic set theories for larger cardinals like Mahlo cardinals.
set-theory large-cardinals
asked Jul 26 at 1:43
Tsavorite Prince
22
closed as unclear what you're asking by Noah Schweber, Shailesh, Xander Henderson, max_zorn, Leucippus Jul 30 at 2:34
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |Â
up vote
-1
down vote
favorite
The title is my question.
I'm curious since I can't seem to find any axiomatic set theory that say that large cardinals exist.
Another thing I’d like to know is that if there are any axiomatic set theories for larger cardinals like Mahlo cardinals.
set-theory large-cardinals
asked Jul 26 at 1:43
Tsavorite Prince
22
closed as unclear what you're asking by Noah Schweber, Shailesh, Xander Henderson, max_zorn, Leucippus Jul 30 at 2:34
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
I don't understand your question. "ZFC+There exists a measurable cardinal" is a set theory that proves the existence of Mahlo cardinals.
– Asaf Karagila
Jul 28 at 23:49
What is an axiomatic set theory for you?
– Andrés E. Caicedo
Jul 29 at 0:21
I will note, though I don't have references offhand, that NFU+Infinity+Choice+Small Ordinals+Large Ordinals is, if I'm recalling correctly, in the neighborhood of ZFC+"there's an $n$-Mahlo cardinal for each $n$". (I believe the result I have in mind comes from Solovay & Holmes.)
– Malice Vidrine
Jul 29 at 2:07
(On further review, that NFU setup might be much stronger than the Mahlo cardinals condition I mention.)
– Malice Vidrine
Jul 29 at 3:56
1
I think for this question to be answered, you need to clarify what you mean by "axiomatic set theory." As Asaf mentioned, "ZFC+a measurable" is an axiomatic set theory proving the existence of a measurable cardinal (and a fortiori of many inaccessibles, many Mahlos, many weakly compacts, ...). It's also worth noting that set theorists routinely work in these theories, so they're not pathological answers either. Or maybe you're asking for set-theoretic axioms which aren't obviously about large cardinals but imply their existence (or at least consistency)?
– Noah Schweber
Jul 29 at 21:49
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
The title is my question.
I'm curious since I can't seem to find any axiomatic set theory that say that large cardinals exist.
Another thing I’d like to know is that if there are any axiomatic set theories for larger cardinals like Mahlo cardinals.
set-theory large-cardinals
asked Jul 26 at 1:43
Tsavorite Prince
22
The title is my question.
I'm curious since I can't seem to find any axiomatic set theory that say that large cardinals exist.
Another thing I’d like to know is that if there are any axiomatic set theories for larger cardinals like Mahlo cardinals.
set-theory large-cardinals
asked Jul 26 at 1:43
Tsavorite Prince
22
edited Jul 28 at 23:39
asked Jul 26 at 1:43
Tsavorite Prince
22
asked Jul 26 at 1:43
Tsavorite Prince
22
asked Jul 26 at 1:43
Tsavorite Prince
22
22
closed as unclear what you're asking by Noah Schweber, Shailesh, Xander Henderson, max_zorn, Leucippus Jul 30 at 2:34
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Noah Schweber, Shailesh, Xander Henderson, max_zorn, Leucippus Jul 30 at 2:34
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
I don't understand your question. "ZFC+There exists a measurable cardinal" is a set theory that proves the existence of Mahlo cardinals.
– Asaf Karagila
Jul 28 at 23:49
What is an axiomatic set theory for you?
– Andrés E. Caicedo
Jul 29 at 0:21
I will note, though I don't have references offhand, that NFU+Infinity+Choice+Small Ordinals+Large Ordinals is, if I'm recalling correctly, in the neighborhood of ZFC+"there's an $n$-Mahlo cardinal for each $n$". (I believe the result I have in mind comes from Solovay & Holmes.)
– Malice Vidrine
Jul 29 at 2:07
(On further review, that NFU setup might be much stronger than the Mahlo cardinals condition I mention.)
– Malice Vidrine
Jul 29 at 3:56
1
I think for this question to be answered, you need to clarify what you mean by "axiomatic set theory." As Asaf mentioned, "ZFC+a measurable" is an axiomatic set theory proving the existence of a measurable cardinal (and a fortiori of many inaccessibles, many Mahlos, many weakly compacts, ...). It's also worth noting that set theorists routinely work in these theories, so they're not pathological answers either. Or maybe you're asking for set-theoretic axioms which aren't obviously about large cardinals but imply their existence (or at least consistency)?
– Noah Schweber
Jul 29 at 21:49
add a comment |Â
3
I don't understand your question. "ZFC+There exists a measurable cardinal" is a set theory that proves the existence of Mahlo cardinals.
– Asaf Karagila
Jul 28 at 23:49
What is an axiomatic set theory for you?
– Andrés E. Caicedo
Jul 29 at 0:21
I will note, though I don't have references offhand, that NFU+Infinity+Choice+Small Ordinals+Large Ordinals is, if I'm recalling correctly, in the neighborhood of ZFC+"there's an $n$-Mahlo cardinal for each $n$". (I believe the result I have in mind comes from Solovay & Holmes.)
– Malice Vidrine
Jul 29 at 2:07
(On further review, that NFU setup might be much stronger than the Mahlo cardinals condition I mention.)
– Malice Vidrine
Jul 29 at 3:56
1
I think for this question to be answered, you need to clarify what you mean by "axiomatic set theory." As Asaf mentioned, "ZFC+a measurable" is an axiomatic set theory proving the existence of a measurable cardinal (and a fortiori of many inaccessibles, many Mahlos, many weakly compacts, ...). It's also worth noting that set theorists routinely work in these theories, so they're not pathological answers either. Or maybe you're asking for set-theoretic axioms which aren't obviously about large cardinals but imply their existence (or at least consistency)?
– Noah Schweber
Jul 29 at 21:49
3
3
I don't understand your question. "ZFC+There exists a measurable cardinal" is a set theory that proves the existence of Mahlo cardinals.
– Asaf Karagila
Jul 28 at 23:49
I don't understand your question. "ZFC+There exists a measurable cardinal" is a set theory that proves the existence of Mahlo cardinals.
– Asaf Karagila
Jul 28 at 23:49
What is an axiomatic set theory for you?
– Andrés E. Caicedo
Jul 29 at 0:21
What is an axiomatic set theory for you?
– Andrés E. Caicedo
Jul 29 at 0:21
I will note, though I don't have references offhand, that NFU+Infinity+Choice+Small Ordinals+Large Ordinals is, if I'm recalling correctly, in the neighborhood of ZFC+"there's an $n$-Mahlo cardinal for each $n$". (I believe the result I have in mind comes from Solovay & Holmes.)
– Malice Vidrine
Jul 29 at 2:07
I will note, though I don't have references offhand, that NFU+Infinity+Choice+Small Ordinals+Large Ordinals is, if I'm recalling correctly, in the neighborhood of ZFC+"there's an $n$-Mahlo cardinal for each $n$". (I believe the result I have in mind comes from Solovay & Holmes.)
– Malice Vidrine
Jul 29 at 2:07
(On further review, that NFU setup might be much stronger than the Mahlo cardinals condition I mention.)
– Malice Vidrine
Jul 29 at 3:56
(On further review, that NFU setup might be much stronger than the Mahlo cardinals condition I mention.)
– Malice Vidrine
Jul 29 at 3:56
1
1
I think for this question to be answered, you need to clarify what you mean by "axiomatic set theory." As Asaf mentioned, "ZFC+a measurable" is an axiomatic set theory proving the existence of a measurable cardinal (and a fortiori of many inaccessibles, many Mahlos, many weakly compacts, ...). It's also worth noting that set theorists routinely work in these theories, so they're not pathological answers either. Or maybe you're asking for set-theoretic axioms which aren't obviously about large cardinals but imply their existence (or at least consistency)?
– Noah Schweber
Jul 29 at 21:49
I think for this question to be answered, you need to clarify what you mean by "axiomatic set theory." As Asaf mentioned, "ZFC+a measurable" is an axiomatic set theory proving the existence of a measurable cardinal (and a fortiori of many inaccessibles, many Mahlos, many weakly compacts, ...). It's also worth noting that set theorists routinely work in these theories, so they're not pathological answers either. Or maybe you're asking for set-theoretic axioms which aren't obviously about large cardinals but imply their existence (or at least consistency)?
– Noah Schweber
Jul 29 at 21:49
add a comment |Â
1 Answer
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One usually takes an axiomatic set theory like $mathrmZFC$ that neither posits their existence nor denies them, and then adjoins a large cardinal axiom $varphi$ of one's choosing, thereby obtaining $mathrmZFC+varphi.$ However there's exceptions, such as TG.
There's also some set theories that posit the existence of collections that "standard" set theories consider "too large" to be sets and consequently disallow, such as MK, and NF.
answered Jul 26 at 1:57
goblin
35.4k1153181
I didn't know that MK, TG, and NF were large cardinal theories.
– Tsavorite Prince
Jul 28 at 17:43
1
MK and NF are not. In fact, the mention of NF is rather puzzling.
– Andrés E. Caicedo
Jul 29 at 0:11
MK does not posit the existence of any large cardinals. It is of higher consistency strength than ZFC, but that's not the same thing. Meanwhile the consistency strength of NF is not known (Holmes has a claimed proof that it is actually quite low - vastly weaker than ZFC - but as far as I know that hasn't been vetted yet).
– Noah Schweber
Jul 29 at 21:46
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
One usually takes an axiomatic set theory like $mathrmZFC$ that neither posits their existence nor denies them, and then adjoins a large cardinal axiom $varphi$ of one's choosing, thereby obtaining $mathrmZFC+varphi.$ However there's exceptions, such as TG.
There's also some set theories that posit the existence of collections that "standard" set theories consider "too large" to be sets and consequently disallow, such as MK, and NF.
answered Jul 26 at 1:57
goblin
35.4k1153181
I didn't know that MK, TG, and NF were large cardinal theories.
– Tsavorite Prince
Jul 28 at 17:43
1
MK and NF are not. In fact, the mention of NF is rather puzzling.
– Andrés E. Caicedo
Jul 29 at 0:11
MK does not posit the existence of any large cardinals. It is of higher consistency strength than ZFC, but that's not the same thing. Meanwhile the consistency strength of NF is not known (Holmes has a claimed proof that it is actually quite low - vastly weaker than ZFC - but as far as I know that hasn't been vetted yet).
– Noah Schweber
Jul 29 at 21:46
add a comment |Â
up vote
4
down vote
One usually takes an axiomatic set theory like $mathrmZFC$ that neither posits their existence nor denies them, and then adjoins a large cardinal axiom $varphi$ of one's choosing, thereby obtaining $mathrmZFC+varphi.$ However there's exceptions, such as TG.
There's also some set theories that posit the existence of collections that "standard" set theories consider "too large" to be sets and consequently disallow, such as MK, and NF.
answered Jul 26 at 1:57
goblin
35.4k1153181
I didn't know that MK, TG, and NF were large cardinal theories.
– Tsavorite Prince
Jul 28 at 17:43
1
MK and NF are not. In fact, the mention of NF is rather puzzling.
– Andrés E. Caicedo
Jul 29 at 0:11
MK does not posit the existence of any large cardinals. It is of higher consistency strength than ZFC, but that's not the same thing. Meanwhile the consistency strength of NF is not known (Holmes has a claimed proof that it is actually quite low - vastly weaker than ZFC - but as far as I know that hasn't been vetted yet).
– Noah Schweber
Jul 29 at 21:46
add a comment |Â
up vote
4
down vote
up vote
4
down vote
One usually takes an axiomatic set theory like $mathrmZFC$ that neither posits their existence nor denies them, and then adjoins a large cardinal axiom $varphi$ of one's choosing, thereby obtaining $mathrmZFC+varphi.$ However there's exceptions, such as TG.
There's also some set theories that posit the existence of collections that "standard" set theories consider "too large" to be sets and consequently disallow, such as MK, and NF.
answered Jul 26 at 1:57
goblin
35.4k1153181
One usually takes an axiomatic set theory like $mathrmZFC$ that neither posits their existence nor denies them, and then adjoins a large cardinal axiom $varphi$ of one's choosing, thereby obtaining $mathrmZFC+varphi.$ However there's exceptions, such as TG.
There's also some set theories that posit the existence of collections that "standard" set theories consider "too large" to be sets and consequently disallow, such as MK, and NF.
answered Jul 26 at 1:57
goblin
35.4k1153181
edited Jul 30 at 1:42
answered Jul 26 at 1:57
goblin
35.4k1153181
answered Jul 26 at 1:57
goblin
35.4k1153181
answered Jul 26 at 1:57
goblin
35.4k1153181
35.4k1153181
I didn't know that MK, TG, and NF were large cardinal theories.
– Tsavorite Prince
Jul 28 at 17:43
1
MK and NF are not. In fact, the mention of NF is rather puzzling.
– Andrés E. Caicedo
Jul 29 at 0:11
MK does not posit the existence of any large cardinals. It is of higher consistency strength than ZFC, but that's not the same thing. Meanwhile the consistency strength of NF is not known (Holmes has a claimed proof that it is actually quite low - vastly weaker than ZFC - but as far as I know that hasn't been vetted yet).
– Noah Schweber
Jul 29 at 21:46
add a comment |Â
I didn't know that MK, TG, and NF were large cardinal theories.
– Tsavorite Prince
Jul 28 at 17:43
1
MK and NF are not. In fact, the mention of NF is rather puzzling.
– Andrés E. Caicedo
Jul 29 at 0:11
MK does not posit the existence of any large cardinals. It is of higher consistency strength than ZFC, but that's not the same thing. Meanwhile the consistency strength of NF is not known (Holmes has a claimed proof that it is actually quite low - vastly weaker than ZFC - but as far as I know that hasn't been vetted yet).
– Noah Schweber
Jul 29 at 21:46
I didn't know that MK, TG, and NF were large cardinal theories.
– Tsavorite Prince
Jul 28 at 17:43
I didn't know that MK, TG, and NF were large cardinal theories.
– Tsavorite Prince
Jul 28 at 17:43
1
1
MK and NF are not. In fact, the mention of NF is rather puzzling.
– Andrés E. Caicedo
Jul 29 at 0:11
MK and NF are not. In fact, the mention of NF is rather puzzling.
– Andrés E. Caicedo
Jul 29 at 0:11
MK does not posit the existence of any large cardinals. It is of higher consistency strength than ZFC, but that's not the same thing. Meanwhile the consistency strength of NF is not known (Holmes has a claimed proof that it is actually quite low - vastly weaker than ZFC - but as far as I know that hasn't been vetted yet).
– Noah Schweber
Jul 29 at 21:46
MK does not posit the existence of any large cardinals. It is of higher consistency strength than ZFC, but that's not the same thing. Meanwhile the consistency strength of NF is not known (Holmes has a claimed proof that it is actually quite low - vastly weaker than ZFC - but as far as I know that hasn't been vetted yet).
– Noah Schweber
Jul 29 at 21:46
add a comment |Â
3
I don't understand your question. "ZFC+There exists a measurable cardinal" is a set theory that proves the existence of Mahlo cardinals.
– Asaf Karagila
Jul 28 at 23:49
What is an axiomatic set theory for you?
– Andrés E. Caicedo
Jul 29 at 0:21
I will note, though I don't have references offhand, that NFU+Infinity+Choice+Small Ordinals+Large Ordinals is, if I'm recalling correctly, in the neighborhood of ZFC+"there's an $n$-Mahlo cardinal for each $n$". (I believe the result I have in mind comes from Solovay & Holmes.)
– Malice Vidrine
Jul 29 at 2:07
(On further review, that NFU setup might be much stronger than the Mahlo cardinals condition I mention.)
– Malice Vidrine
Jul 29 at 3:56
1
I think for this question to be answered, you need to clarify what you mean by "axiomatic set theory." As Asaf mentioned, "ZFC+a measurable" is an axiomatic set theory proving the existence of a measurable cardinal (and a fortiori of many inaccessibles, many Mahlos, many weakly compacts, ...). It's also worth noting that set theorists routinely work in these theories, so they're not pathological answers either. Or maybe you're asking for set-theoretic axioms which aren't obviously about large cardinals but imply their existence (or at least consistency)?
– Noah Schweber
Jul 29 at 21:49