Mixing
Is Godel's incompleteness theorem unavoidable?
Clash Royale CLAN TAG#URR8PPP
up vote
3
down vote
favorite
So after Godel's Incompleteness theorem and the fact that some theorems mathematicians are interested in are independent of ZFC (e.g. Continuum Hypothesis) is there some hope for some other foundational theory which will be (provably) complete and the fact that it is complete will not imply it`s inconsistency ? Furthermore, is there hope for such theory to have have most of axioms to be intuitive, I mean not to be ad hoc ? If it is not, can it still be that there is some way different from foundations based on one foundating theory from which we build all other theories, which will allow us to avoid Godel incompletness theorem ? Or is it just unavoidable ?
soft-question foundations incompleteness
edited 19 hours ago
TheSimpliFire
9,18651651
asked 19 hours ago
îріù ïрþш
964412
add a comment |Â
up vote
3
down vote
favorite
So after Godel's Incompleteness theorem and the fact that some theorems mathematicians are interested in are independent of ZFC (e.g. Continuum Hypothesis) is there some hope for some other foundational theory which will be (provably) complete and the fact that it is complete will not imply it`s inconsistency ? Furthermore, is there hope for such theory to have have most of axioms to be intuitive, I mean not to be ad hoc ? If it is not, can it still be that there is some way different from foundations based on one foundating theory from which we build all other theories, which will allow us to avoid Godel incompletness theorem ? Or is it just unavoidable ?
soft-question foundations incompleteness
edited 19 hours ago
TheSimpliFire
9,18651651
asked 19 hours ago
îріù ïрþш
964412
1
You can avoid them in paraconsistent logics.
– Shaun
19 hours ago
3
The incompleteness theorem applies, roughly speaking, to any collection of axioms which 1) can be described by a computer program and 2) is powerful enough to describe computer programs; this is the meat of what's necessary to run the proof. If your axioms don't satisfy 1) then in general you have no way of telling what a valid proof from your axioms looks like. And if they don't satisfy 2) then your axioms are not powerful enough to do mathematics. See scottaaronson.com/democritus/lec3.html for some details.
– Qiaochu Yuan
18 hours ago
I think Qiaochu's comment says it all : Gödel's theorem essentially arises from the tension between being powerful enough to describe what's interesting to us, and being weak enough as to be interesting to us (a non recursively enumerable theory could be awesome, but it wouldn't be interesting to any user)
– Max
17 hours ago
The problem (as I see it) is that in order to ask questions about an infinite set we have to approach it from a finite perspective, which obviously is going to come short. If we could do uncountable computations, we could simply check the continuum hypothesis by checking the cardinality of every subset of $mathbbN$.
– Dark Malthorp
16 hours ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
So after Godel's Incompleteness theorem and the fact that some theorems mathematicians are interested in are independent of ZFC (e.g. Continuum Hypothesis) is there some hope for some other foundational theory which will be (provably) complete and the fact that it is complete will not imply it`s inconsistency ? Furthermore, is there hope for such theory to have have most of axioms to be intuitive, I mean not to be ad hoc ? If it is not, can it still be that there is some way different from foundations based on one foundating theory from which we build all other theories, which will allow us to avoid Godel incompletness theorem ? Or is it just unavoidable ?
soft-question foundations incompleteness
edited 19 hours ago
TheSimpliFire
9,18651651
asked 19 hours ago
îріù ïрþш
964412
So after Godel's Incompleteness theorem and the fact that some theorems mathematicians are interested in are independent of ZFC (e.g. Continuum Hypothesis) is there some hope for some other foundational theory which will be (provably) complete and the fact that it is complete will not imply it`s inconsistency ? Furthermore, is there hope for such theory to have have most of axioms to be intuitive, I mean not to be ad hoc ? If it is not, can it still be that there is some way different from foundations based on one foundating theory from which we build all other theories, which will allow us to avoid Godel incompletness theorem ? Or is it just unavoidable ?
soft-question foundations incompleteness
edited 19 hours ago
TheSimpliFire
9,18651651
asked 19 hours ago
îріù ïрþш
964412
edited 19 hours ago
TheSimpliFire
9,18651651
edited 19 hours ago
TheSimpliFire
9,18651651
edited 19 hours ago
TheSimpliFire
9,18651651
9,18651651
asked 19 hours ago
îріù ïрþш
964412
asked 19 hours ago
îріù ïрþш
964412
asked 19 hours ago
îріù ïрþш
964412
964412
1
You can avoid them in paraconsistent logics.
– Shaun
19 hours ago
3
The incompleteness theorem applies, roughly speaking, to any collection of axioms which 1) can be described by a computer program and 2) is powerful enough to describe computer programs; this is the meat of what's necessary to run the proof. If your axioms don't satisfy 1) then in general you have no way of telling what a valid proof from your axioms looks like. And if they don't satisfy 2) then your axioms are not powerful enough to do mathematics. See scottaaronson.com/democritus/lec3.html for some details.
– Qiaochu Yuan
18 hours ago
I think Qiaochu's comment says it all : Gödel's theorem essentially arises from the tension between being powerful enough to describe what's interesting to us, and being weak enough as to be interesting to us (a non recursively enumerable theory could be awesome, but it wouldn't be interesting to any user)
– Max
17 hours ago
The problem (as I see it) is that in order to ask questions about an infinite set we have to approach it from a finite perspective, which obviously is going to come short. If we could do uncountable computations, we could simply check the continuum hypothesis by checking the cardinality of every subset of $mathbbN$.
– Dark Malthorp
16 hours ago
add a comment |Â
1
You can avoid them in paraconsistent logics.
– Shaun
19 hours ago
3
The incompleteness theorem applies, roughly speaking, to any collection of axioms which 1) can be described by a computer program and 2) is powerful enough to describe computer programs; this is the meat of what's necessary to run the proof. If your axioms don't satisfy 1) then in general you have no way of telling what a valid proof from your axioms looks like. And if they don't satisfy 2) then your axioms are not powerful enough to do mathematics. See scottaaronson.com/democritus/lec3.html for some details.
– Qiaochu Yuan
18 hours ago
I think Qiaochu's comment says it all : Gödel's theorem essentially arises from the tension between being powerful enough to describe what's interesting to us, and being weak enough as to be interesting to us (a non recursively enumerable theory could be awesome, but it wouldn't be interesting to any user)
– Max
17 hours ago
The problem (as I see it) is that in order to ask questions about an infinite set we have to approach it from a finite perspective, which obviously is going to come short. If we could do uncountable computations, we could simply check the continuum hypothesis by checking the cardinality of every subset of $mathbbN$.
– Dark Malthorp
16 hours ago
1
1
You can avoid them in paraconsistent logics.
– Shaun
19 hours ago
You can avoid them in paraconsistent logics.
– Shaun
19 hours ago
3
3
The incompleteness theorem applies, roughly speaking, to any collection of axioms which 1) can be described by a computer program and 2) is powerful enough to describe computer programs; this is the meat of what's necessary to run the proof. If your axioms don't satisfy 1) then in general you have no way of telling what a valid proof from your axioms looks like. And if they don't satisfy 2) then your axioms are not powerful enough to do mathematics. See scottaaronson.com/democritus/lec3.html for some details.
– Qiaochu Yuan
18 hours ago
The incompleteness theorem applies, roughly speaking, to any collection of axioms which 1) can be described by a computer program and 2) is powerful enough to describe computer programs; this is the meat of what's necessary to run the proof. If your axioms don't satisfy 1) then in general you have no way of telling what a valid proof from your axioms looks like. And if they don't satisfy 2) then your axioms are not powerful enough to do mathematics. See scottaaronson.com/democritus/lec3.html for some details.
– Qiaochu Yuan
18 hours ago
I think Qiaochu's comment says it all : Gödel's theorem essentially arises from the tension between being powerful enough to describe what's interesting to us, and being weak enough as to be interesting to us (a non recursively enumerable theory could be awesome, but it wouldn't be interesting to any user)
– Max
17 hours ago
I think Qiaochu's comment says it all : Gödel's theorem essentially arises from the tension between being powerful enough to describe what's interesting to us, and being weak enough as to be interesting to us (a non recursively enumerable theory could be awesome, but it wouldn't be interesting to any user)
– Max
17 hours ago
The problem (as I see it) is that in order to ask questions about an infinite set we have to approach it from a finite perspective, which obviously is going to come short. If we could do uncountable computations, we could simply check the continuum hypothesis by checking the cardinality of every subset of $mathbbN$.
– Dark Malthorp
16 hours ago
The problem (as I see it) is that in order to ask questions about an infinite set we have to approach it from a finite perspective, which obviously is going to come short. If we could do uncountable computations, we could simply check the continuum hypothesis by checking the cardinality of every subset of $mathbbN$.
– Dark Malthorp
16 hours ago
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2872957%2fis-godels-incompleteness-theorem-unavoidable%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
You can avoid them in paraconsistent logics.
– Shaun
19 hours ago
3
The incompleteness theorem applies, roughly speaking, to any collection of axioms which 1) can be described by a computer program and 2) is powerful enough to describe computer programs; this is the meat of what's necessary to run the proof. If your axioms don't satisfy 1) then in general you have no way of telling what a valid proof from your axioms looks like. And if they don't satisfy 2) then your axioms are not powerful enough to do mathematics. See scottaaronson.com/democritus/lec3.html for some details.
– Qiaochu Yuan
18 hours ago
I think Qiaochu's comment says it all : Gödel's theorem essentially arises from the tension between being powerful enough to describe what's interesting to us, and being weak enough as to be interesting to us (a non recursively enumerable theory could be awesome, but it wouldn't be interesting to any user)
– Max
17 hours ago
The problem (as I see it) is that in order to ask questions about an infinite set we have to approach it from a finite perspective, which obviously is going to come short. If we could do uncountable computations, we could simply check the continuum hypothesis by checking the cardinality of every subset of $mathbbN$.
– Dark Malthorp
16 hours ago