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Assumptions and axioms [closed]
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By Godel, if mathematics is consistent it must be incomplete. Given that mathematics is essentially believed to be a tautology it must be incomplete.
Can we assume mathematics itself as an axiom? What are the limits on scaling up axioms? Do they have to be bare bones axioms or can we take something as large as mathematics as an entire axiom?
Thanks.
logic soft-question axioms
asked Aug 6 at 19:02
George Thomas
51416
closed as unclear what you're asking by Carl Mummert, Simply Beautiful Art, Xander Henderson, amWhy, Leucippus Aug 7 at 1:09
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.
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up vote
-5
down vote
favorite
By Godel, if mathematics is consistent it must be incomplete. Given that mathematics is essentially believed to be a tautology it must be incomplete.
Can we assume mathematics itself as an axiom? What are the limits on scaling up axioms? Do they have to be bare bones axioms or can we take something as large as mathematics as an entire axiom?
Thanks.
logic soft-question axioms
asked Aug 6 at 19:02
George Thomas
51416
closed as unclear what you're asking by Carl Mummert, Simply Beautiful Art, Xander Henderson, amWhy, Leucippus Aug 7 at 1:09
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.
10
Uh, what???????
– Asaf Karagila♦
Aug 6 at 19:04
1
I guess in your question there is a huge confusion about crucial notions of logic and in particular about Gödel's incompleteness theorem. I suggest to start reading Wikipedia's page on incompleteness theorem, then reading Nagel's and Newman's Godel's proof.
– Taroccoesbrocco
Aug 6 at 19:11
add a comment |Â
up vote
-5
down vote
favorite
up vote
-5
down vote
favorite
By Godel, if mathematics is consistent it must be incomplete. Given that mathematics is essentially believed to be a tautology it must be incomplete.
Can we assume mathematics itself as an axiom? What are the limits on scaling up axioms? Do they have to be bare bones axioms or can we take something as large as mathematics as an entire axiom?
Thanks.
logic soft-question axioms
asked Aug 6 at 19:02
George Thomas
51416
By Godel, if mathematics is consistent it must be incomplete. Given that mathematics is essentially believed to be a tautology it must be incomplete.
Can we assume mathematics itself as an axiom? What are the limits on scaling up axioms? Do they have to be bare bones axioms or can we take something as large as mathematics as an entire axiom?
Thanks.
logic soft-question axioms
asked Aug 6 at 19:02
George Thomas
51416
asked Aug 6 at 19:02
George Thomas
51416
asked Aug 6 at 19:02
George Thomas
51416
asked Aug 6 at 19:02
George Thomas
51416
51416
closed as unclear what you're asking by Carl Mummert, Simply Beautiful Art, Xander Henderson, amWhy, Leucippus Aug 7 at 1:09
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 Carl Mummert, Simply Beautiful Art, Xander Henderson, amWhy, Leucippus Aug 7 at 1:09
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.
10
Uh, what???????
– Asaf Karagila♦
Aug 6 at 19:04
1
I guess in your question there is a huge confusion about crucial notions of logic and in particular about Gödel's incompleteness theorem. I suggest to start reading Wikipedia's page on incompleteness theorem, then reading Nagel's and Newman's Godel's proof.
– Taroccoesbrocco
Aug 6 at 19:11
add a comment |Â
10
Uh, what???????
– Asaf Karagila♦
Aug 6 at 19:04
1
I guess in your question there is a huge confusion about crucial notions of logic and in particular about Gödel's incompleteness theorem. I suggest to start reading Wikipedia's page on incompleteness theorem, then reading Nagel's and Newman's Godel's proof.
– Taroccoesbrocco
Aug 6 at 19:11
10
10
Uh, what???????
– Asaf Karagila♦
Aug 6 at 19:04
Uh, what???????
– Asaf Karagila♦
Aug 6 at 19:04
1
1
I guess in your question there is a huge confusion about crucial notions of logic and in particular about Gödel's incompleteness theorem. I suggest to start reading Wikipedia's page on incompleteness theorem, then reading Nagel's and Newman's Godel's proof.
– Taroccoesbrocco
Aug 6 at 19:11
I guess in your question there is a huge confusion about crucial notions of logic and in particular about Gödel's incompleteness theorem. I suggest to start reading Wikipedia's page on incompleteness theorem, then reading Nagel's and Newman's Godel's proof.
– Taroccoesbrocco
Aug 6 at 19:11
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
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accepted
Mathematics is not generally believed to be a tautology. There are some who regard any given mathematical theorem as being "true by definition" but this is not the same as saying mathematics itself is a tautology, whatever that might mean. On another note something has to be a statement before it can or cannot be a tautology and "mathematics" does not qualify as a statement.
answered Aug 6 at 19:10
Daron
4,3581923
excellent answer thank you!
– George Thomas
Aug 6 at 19:11
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Mathematics is not generally believed to be a tautology. There are some who regard any given mathematical theorem as being "true by definition" but this is not the same as saying mathematics itself is a tautology, whatever that might mean. On another note something has to be a statement before it can or cannot be a tautology and "mathematics" does not qualify as a statement.
answered Aug 6 at 19:10
Daron
4,3581923
excellent answer thank you!
– George Thomas
Aug 6 at 19:11
add a comment |Â
up vote
1
down vote
accepted
Mathematics is not generally believed to be a tautology. There are some who regard any given mathematical theorem as being "true by definition" but this is not the same as saying mathematics itself is a tautology, whatever that might mean. On another note something has to be a statement before it can or cannot be a tautology and "mathematics" does not qualify as a statement.
answered Aug 6 at 19:10
Daron
4,3581923
excellent answer thank you!
– George Thomas
Aug 6 at 19:11
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Mathematics is not generally believed to be a tautology. There are some who regard any given mathematical theorem as being "true by definition" but this is not the same as saying mathematics itself is a tautology, whatever that might mean. On another note something has to be a statement before it can or cannot be a tautology and "mathematics" does not qualify as a statement.
answered Aug 6 at 19:10
Daron
4,3581923
Mathematics is not generally believed to be a tautology. There are some who regard any given mathematical theorem as being "true by definition" but this is not the same as saying mathematics itself is a tautology, whatever that might mean. On another note something has to be a statement before it can or cannot be a tautology and "mathematics" does not qualify as a statement.
answered Aug 6 at 19:10
Daron
4,3581923
answered Aug 6 at 19:10
Daron
4,3581923
answered Aug 6 at 19:10
Daron
4,3581923
answered Aug 6 at 19:10
Daron
4,3581923
4,3581923
excellent answer thank you!
– George Thomas
Aug 6 at 19:11
add a comment |Â
excellent answer thank you!
– George Thomas
Aug 6 at 19:11
excellent answer thank you!
– George Thomas
Aug 6 at 19:11
excellent answer thank you!
– George Thomas
Aug 6 at 19:11
add a comment |Â
10
Uh, what???????
– Asaf Karagila♦
Aug 6 at 19:04
1
I guess in your question there is a huge confusion about crucial notions of logic and in particular about Gödel's incompleteness theorem. I suggest to start reading Wikipedia's page on incompleteness theorem, then reading Nagel's and Newman's Godel's proof.
– Taroccoesbrocco
Aug 6 at 19:11