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Why is the Hilbert-Bernays paradox paradoxical?
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The Hilbert-Bernays Paradox is produced by defining h as '(the referent of h) + 1'. Why is this a paradox? It seems strange to believe that we could define h in terms of itself. I suspect I'm missing some context, but I can't find anything else about this paradox online that isn't pay-walled.
paradoxes
asked Jul 18 at 0:40
Casebash
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up vote
4
down vote
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The Hilbert-Bernays Paradox is produced by defining h as '(the referent of h) + 1'. Why is this a paradox? It seems strange to believe that we could define h in terms of itself. I suspect I'm missing some context, but I can't find anything else about this paradox online that isn't pay-walled.
paradoxes
asked Jul 18 at 0:40
Casebash
5,50634070
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
The Hilbert-Bernays Paradox is produced by defining h as '(the referent of h) + 1'. Why is this a paradox? It seems strange to believe that we could define h in terms of itself. I suspect I'm missing some context, but I can't find anything else about this paradox online that isn't pay-walled.
paradoxes
asked Jul 18 at 0:40
Casebash
5,50634070
The Hilbert-Bernays Paradox is produced by defining h as '(the referent of h) + 1'. Why is this a paradox? It seems strange to believe that we could define h in terms of itself. I suspect I'm missing some context, but I can't find anything else about this paradox online that isn't pay-walled.
paradoxes
asked Jul 18 at 0:40
Casebash
5,50634070
edited Jul 20 at 8:22
asked Jul 18 at 0:40
Casebash
5,50634070
asked Jul 18 at 0:40
Casebash
5,50634070
asked Jul 18 at 0:40
Casebash
5,50634070
5,50634070
add a comment |Â
add a comment |Â
1 Answer
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up vote
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I find the wiki article somewhat confusing. I don't have my copy of the Grundlagen at hand at the moment, but I think the wiki article has actually botched the phrasing of the paragraph. Below I'll first re-summarize the Liar (just for completeness and clarity), give my gloss on the HBG, and end by saying a bit about how reference and truth are related.
First, let's go back to something well-known. The classical Liar Paradox
$(*)quad$This statement is false
shows that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internal) truth definition. If we have self-reference and a definition of truth, we can formula $(*)$, and classical logic then leads us to inconsistency since we can deduce both $(*)$ and $neg(*)$. Phrased this way, the Liar isn't really a paradox, it's a theorem about the limitations of logical systems. In light of the diagonal lemma, which permits (certain kinds of) self-reference in arithmetic, the Liar yields Tarski's theorem on the undefinability of truth; with an additional twist (replacing "truth" with "provability" - fine, and then applying an additional technical trick), we get Godel's first incompleteness theorem.
OK, now what about the Hilbert-Bernays Paradox? As far as I can tell, as phrased by the wiki article it's not actually a problem, but it can be easily modified to be nasty:
$(sharp)quad$ $1+n$, if the referent of this expression is $n$, and $0$ if this expression has no referent.
Note this second clause: merely saying "Oh, $(sharp)$ doesn't have a referent" doesn't save us now! What we're looking at is very reminiscent of the Liar, $(*)$, above. This time it yields the following theorem: that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internally) definable "referent functor" (= a way to say "the referent of ---" in the internal language).
My own instinct - and I think I'm not too rare in this - is to be suspicious of the role of self-reference. But just like in the case of the Liar, the diagonal lemma tells us that (in arithmetic at least) we actually do have enough self-referential ability to get into hot water! So in "reasonable" logical contexts (= nontrivial classically-founded computably-axiomatizable theories strong enough to support arithmetic), we conclude that the problem lies with defining "reference" (just as the Liar, via Tarski, showed that the real problem in arithmetic is defining "truth").
Note that truth and reference are (unsurprisingly) closely related in the context of arithmetic:
Given a sentence $varphi$, consider the expression "$0$ if $varphi$, $1$ if not $varphi$;" if we know the referent for this expression, then we know whether $varphi$ is true or not.
Given a formula $p(x)$, consider the sentences $p(0)$, $p(1)$, $p(2)$, $p(3)$, ... Exactly one of these is true, and if we know which is true then we know the referent of the expression "the unique $x$ satisfying $p$."
In general, reference is more powerful than truth, however, since the domain we live in may not consist only of definable elements (so we won't be able to "search through it" for the referent as we did in the second bulletpoint above). Given this, we can actually say that in a sense $(sharp)$ is "more basic" than $(*)$.
$^*$It's worth observing that in weaker logical systems, a single inconsistency doesn't necessarily lead to explosion. If we don't insist on classical logic, we can have nontrivial logical systems which have inconsistencies. The relevant term here is paraconsistency, and one response to the various semantic paradoxes like the Liar is to work with them, inside a paraconsistent system.
answered Jul 18 at 1:50
Noah Schweber
111k9140262
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
accepted
I find the wiki article somewhat confusing. I don't have my copy of the Grundlagen at hand at the moment, but I think the wiki article has actually botched the phrasing of the paragraph. Below I'll first re-summarize the Liar (just for completeness and clarity), give my gloss on the HBG, and end by saying a bit about how reference and truth are related.
First, let's go back to something well-known. The classical Liar Paradox
$(*)quad$This statement is false
shows that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internal) truth definition. If we have self-reference and a definition of truth, we can formula $(*)$, and classical logic then leads us to inconsistency since we can deduce both $(*)$ and $neg(*)$. Phrased this way, the Liar isn't really a paradox, it's a theorem about the limitations of logical systems. In light of the diagonal lemma, which permits (certain kinds of) self-reference in arithmetic, the Liar yields Tarski's theorem on the undefinability of truth; with an additional twist (replacing "truth" with "provability" - fine, and then applying an additional technical trick), we get Godel's first incompleteness theorem.
OK, now what about the Hilbert-Bernays Paradox? As far as I can tell, as phrased by the wiki article it's not actually a problem, but it can be easily modified to be nasty:
$(sharp)quad$ $1+n$, if the referent of this expression is $n$, and $0$ if this expression has no referent.
Note this second clause: merely saying "Oh, $(sharp)$ doesn't have a referent" doesn't save us now! What we're looking at is very reminiscent of the Liar, $(*)$, above. This time it yields the following theorem: that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internally) definable "referent functor" (= a way to say "the referent of ---" in the internal language).
My own instinct - and I think I'm not too rare in this - is to be suspicious of the role of self-reference. But just like in the case of the Liar, the diagonal lemma tells us that (in arithmetic at least) we actually do have enough self-referential ability to get into hot water! So in "reasonable" logical contexts (= nontrivial classically-founded computably-axiomatizable theories strong enough to support arithmetic), we conclude that the problem lies with defining "reference" (just as the Liar, via Tarski, showed that the real problem in arithmetic is defining "truth").
Note that truth and reference are (unsurprisingly) closely related in the context of arithmetic:
Given a sentence $varphi$, consider the expression "$0$ if $varphi$, $1$ if not $varphi$;" if we know the referent for this expression, then we know whether $varphi$ is true or not.
Given a formula $p(x)$, consider the sentences $p(0)$, $p(1)$, $p(2)$, $p(3)$, ... Exactly one of these is true, and if we know which is true then we know the referent of the expression "the unique $x$ satisfying $p$."
In general, reference is more powerful than truth, however, since the domain we live in may not consist only of definable elements (so we won't be able to "search through it" for the referent as we did in the second bulletpoint above). Given this, we can actually say that in a sense $(sharp)$ is "more basic" than $(*)$.
$^*$It's worth observing that in weaker logical systems, a single inconsistency doesn't necessarily lead to explosion. If we don't insist on classical logic, we can have nontrivial logical systems which have inconsistencies. The relevant term here is paraconsistency, and one response to the various semantic paradoxes like the Liar is to work with them, inside a paraconsistent system.
answered Jul 18 at 1:50
Noah Schweber
111k9140262
add a comment |Â
up vote
4
down vote
accepted
I find the wiki article somewhat confusing. I don't have my copy of the Grundlagen at hand at the moment, but I think the wiki article has actually botched the phrasing of the paragraph. Below I'll first re-summarize the Liar (just for completeness and clarity), give my gloss on the HBG, and end by saying a bit about how reference and truth are related.
First, let's go back to something well-known. The classical Liar Paradox
$(*)quad$This statement is false
shows that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internal) truth definition. If we have self-reference and a definition of truth, we can formula $(*)$, and classical logic then leads us to inconsistency since we can deduce both $(*)$ and $neg(*)$. Phrased this way, the Liar isn't really a paradox, it's a theorem about the limitations of logical systems. In light of the diagonal lemma, which permits (certain kinds of) self-reference in arithmetic, the Liar yields Tarski's theorem on the undefinability of truth; with an additional twist (replacing "truth" with "provability" - fine, and then applying an additional technical trick), we get Godel's first incompleteness theorem.
OK, now what about the Hilbert-Bernays Paradox? As far as I can tell, as phrased by the wiki article it's not actually a problem, but it can be easily modified to be nasty:
$(sharp)quad$ $1+n$, if the referent of this expression is $n$, and $0$ if this expression has no referent.
Note this second clause: merely saying "Oh, $(sharp)$ doesn't have a referent" doesn't save us now! What we're looking at is very reminiscent of the Liar, $(*)$, above. This time it yields the following theorem: that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internally) definable "referent functor" (= a way to say "the referent of ---" in the internal language).
My own instinct - and I think I'm not too rare in this - is to be suspicious of the role of self-reference. But just like in the case of the Liar, the diagonal lemma tells us that (in arithmetic at least) we actually do have enough self-referential ability to get into hot water! So in "reasonable" logical contexts (= nontrivial classically-founded computably-axiomatizable theories strong enough to support arithmetic), we conclude that the problem lies with defining "reference" (just as the Liar, via Tarski, showed that the real problem in arithmetic is defining "truth").
Note that truth and reference are (unsurprisingly) closely related in the context of arithmetic:
Given a sentence $varphi$, consider the expression "$0$ if $varphi$, $1$ if not $varphi$;" if we know the referent for this expression, then we know whether $varphi$ is true or not.
Given a formula $p(x)$, consider the sentences $p(0)$, $p(1)$, $p(2)$, $p(3)$, ... Exactly one of these is true, and if we know which is true then we know the referent of the expression "the unique $x$ satisfying $p$."
In general, reference is more powerful than truth, however, since the domain we live in may not consist only of definable elements (so we won't be able to "search through it" for the referent as we did in the second bulletpoint above). Given this, we can actually say that in a sense $(sharp)$ is "more basic" than $(*)$.
$^*$It's worth observing that in weaker logical systems, a single inconsistency doesn't necessarily lead to explosion. If we don't insist on classical logic, we can have nontrivial logical systems which have inconsistencies. The relevant term here is paraconsistency, and one response to the various semantic paradoxes like the Liar is to work with them, inside a paraconsistent system.
answered Jul 18 at 1:50
Noah Schweber
111k9140262
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
I find the wiki article somewhat confusing. I don't have my copy of the Grundlagen at hand at the moment, but I think the wiki article has actually botched the phrasing of the paragraph. Below I'll first re-summarize the Liar (just for completeness and clarity), give my gloss on the HBG, and end by saying a bit about how reference and truth are related.
First, let's go back to something well-known. The classical Liar Paradox
$(*)quad$This statement is false
shows that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internal) truth definition. If we have self-reference and a definition of truth, we can formula $(*)$, and classical logic then leads us to inconsistency since we can deduce both $(*)$ and $neg(*)$. Phrased this way, the Liar isn't really a paradox, it's a theorem about the limitations of logical systems. In light of the diagonal lemma, which permits (certain kinds of) self-reference in arithmetic, the Liar yields Tarski's theorem on the undefinability of truth; with an additional twist (replacing "truth" with "provability" - fine, and then applying an additional technical trick), we get Godel's first incompleteness theorem.
OK, now what about the Hilbert-Bernays Paradox? As far as I can tell, as phrased by the wiki article it's not actually a problem, but it can be easily modified to be nasty:
$(sharp)quad$ $1+n$, if the referent of this expression is $n$, and $0$ if this expression has no referent.
Note this second clause: merely saying "Oh, $(sharp)$ doesn't have a referent" doesn't save us now! What we're looking at is very reminiscent of the Liar, $(*)$, above. This time it yields the following theorem: that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internally) definable "referent functor" (= a way to say "the referent of ---" in the internal language).
My own instinct - and I think I'm not too rare in this - is to be suspicious of the role of self-reference. But just like in the case of the Liar, the diagonal lemma tells us that (in arithmetic at least) we actually do have enough self-referential ability to get into hot water! So in "reasonable" logical contexts (= nontrivial classically-founded computably-axiomatizable theories strong enough to support arithmetic), we conclude that the problem lies with defining "reference" (just as the Liar, via Tarski, showed that the real problem in arithmetic is defining "truth").
Note that truth and reference are (unsurprisingly) closely related in the context of arithmetic:
Given a sentence $varphi$, consider the expression "$0$ if $varphi$, $1$ if not $varphi$;" if we know the referent for this expression, then we know whether $varphi$ is true or not.
Given a formula $p(x)$, consider the sentences $p(0)$, $p(1)$, $p(2)$, $p(3)$, ... Exactly one of these is true, and if we know which is true then we know the referent of the expression "the unique $x$ satisfying $p$."
In general, reference is more powerful than truth, however, since the domain we live in may not consist only of definable elements (so we won't be able to "search through it" for the referent as we did in the second bulletpoint above). Given this, we can actually say that in a sense $(sharp)$ is "more basic" than $(*)$.
$^*$It's worth observing that in weaker logical systems, a single inconsistency doesn't necessarily lead to explosion. If we don't insist on classical logic, we can have nontrivial logical systems which have inconsistencies. The relevant term here is paraconsistency, and one response to the various semantic paradoxes like the Liar is to work with them, inside a paraconsistent system.
answered Jul 18 at 1:50
Noah Schweber
111k9140262
I find the wiki article somewhat confusing. I don't have my copy of the Grundlagen at hand at the moment, but I think the wiki article has actually botched the phrasing of the paragraph. Below I'll first re-summarize the Liar (just for completeness and clarity), give my gloss on the HBG, and end by saying a bit about how reference and truth are related.
First, let's go back to something well-known. The classical Liar Paradox
$(*)quad$This statement is false
shows that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internal) truth definition. If we have self-reference and a definition of truth, we can formula $(*)$, and classical logic then leads us to inconsistency since we can deduce both $(*)$ and $neg(*)$. Phrased this way, the Liar isn't really a paradox, it's a theorem about the limitations of logical systems. In light of the diagonal lemma, which permits (certain kinds of) self-reference in arithmetic, the Liar yields Tarski's theorem on the undefinability of truth; with an additional twist (replacing "truth" with "provability" - fine, and then applying an additional technical trick), we get Godel's first incompleteness theorem.
OK, now what about the Hilbert-Bernays Paradox? As far as I can tell, as phrased by the wiki article it's not actually a problem, but it can be easily modified to be nasty:
$(sharp)quad$ $1+n$, if the referent of this expression is $n$, and $0$ if this expression has no referent.
Note this second clause: merely saying "Oh, $(sharp)$ doesn't have a referent" doesn't save us now! What we're looking at is very reminiscent of the Liar, $(*)$, above. This time it yields the following theorem: that we cannot sustain classical logic, consistency (= non-triviality, given$^*$ classical logic), self-reference, and a(n internally) definable "referent functor" (= a way to say "the referent of ---" in the internal language).
My own instinct - and I think I'm not too rare in this - is to be suspicious of the role of self-reference. But just like in the case of the Liar, the diagonal lemma tells us that (in arithmetic at least) we actually do have enough self-referential ability to get into hot water! So in "reasonable" logical contexts (= nontrivial classically-founded computably-axiomatizable theories strong enough to support arithmetic), we conclude that the problem lies with defining "reference" (just as the Liar, via Tarski, showed that the real problem in arithmetic is defining "truth").
Note that truth and reference are (unsurprisingly) closely related in the context of arithmetic:
Given a sentence $varphi$, consider the expression "$0$ if $varphi$, $1$ if not $varphi$;" if we know the referent for this expression, then we know whether $varphi$ is true or not.
Given a formula $p(x)$, consider the sentences $p(0)$, $p(1)$, $p(2)$, $p(3)$, ... Exactly one of these is true, and if we know which is true then we know the referent of the expression "the unique $x$ satisfying $p$."
In general, reference is more powerful than truth, however, since the domain we live in may not consist only of definable elements (so we won't be able to "search through it" for the referent as we did in the second bulletpoint above). Given this, we can actually say that in a sense $(sharp)$ is "more basic" than $(*)$.
$^*$It's worth observing that in weaker logical systems, a single inconsistency doesn't necessarily lead to explosion. If we don't insist on classical logic, we can have nontrivial logical systems which have inconsistencies. The relevant term here is paraconsistency, and one response to the various semantic paradoxes like the Liar is to work with them, inside a paraconsistent system.
answered Jul 18 at 1:50
Noah Schweber
111k9140262
edited Jul 18 at 2:00
answered Jul 18 at 1:50
Noah Schweber
111k9140262
answered Jul 18 at 1:50
Noah Schweber
111k9140262
answered Jul 18 at 1:50
Noah Schweber
111k9140262
111k9140262
add a comment |Â
add a comment |Â
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