Questions in proof theory (interpretation of PRA in PA, Girards book from '87)
I am working through the above mentioned book, 'proof theory and logical complexity, volume 1', with some trouble here and there.
I would be glad if someone can help me with some of the exercises, clarify things when I can't work out the sense/meaning or help with the understanding of the proofs.
This is about example 1.4.6, I really have major problems here.
1) The author uses the word 'containing' in the first line. Apparently here he does not have in mind the definition of containment which comes RIGHT BEFORE this example, but instead he just means 'the axioms of EA and induction'. Right?
2) I don't see how to use theorem 1.3.5 to prove that PA contains PRA. (Here he uses the word 'contains' in accordance with his definition.)
3) I don't see what role is played by the chinese remainder theorem. I don't need to see a proof of it, and of course I can look it up online, but I don't see how it is used here, in this case, what exact formulation would be necessary etc.
4) What about the formula Prm(x,i,z)? What if x does NOT encode a sequence or i>=n. Then I guess the formula is supposed to be true, right?`
5) I have no idea what the long formula after 'The interpretation of $A_{Rfg}$ is' is saying and it is to me not at all 'immediate' that the interpretations of axioms (ix) and (x) hold.
6) When he tells why PA is prim. rec., in the end, why does he skip axiom (xv) of EA? Wouldn't it be necessary to include it?
Thanks a lot for your help,
Regards,
Ettore
logic proof-theory meta-math
edited Nov 29 '18 at 11:08
Mauro ALLEGRANZA
64.5k448112
asked Nov 29 '18 at 10:51
Ettore
969
add a comment |
I am working through the above mentioned book, 'proof theory and logical complexity, volume 1', with some trouble here and there.
I would be glad if someone can help me with some of the exercises, clarify things when I can't work out the sense/meaning or help with the understanding of the proofs.
This is about example 1.4.6, I really have major problems here.
1) The author uses the word 'containing' in the first line. Apparently here he does not have in mind the definition of containment which comes RIGHT BEFORE this example, but instead he just means 'the axioms of EA and induction'. Right?
2) I don't see how to use theorem 1.3.5 to prove that PA contains PRA. (Here he uses the word 'contains' in accordance with his definition.)
3) I don't see what role is played by the chinese remainder theorem. I don't need to see a proof of it, and of course I can look it up online, but I don't see how it is used here, in this case, what exact formulation would be necessary etc.
4) What about the formula Prm(x,i,z)? What if x does NOT encode a sequence or i>=n. Then I guess the formula is supposed to be true, right?`
5) I have no idea what the long formula after 'The interpretation of $A_{Rfg}$ is' is saying and it is to me not at all 'immediate' that the interpretations of axioms (ix) and (x) hold.
6) When he tells why PA is prim. rec., in the end, why does he skip axiom (xv) of EA? Wouldn't it be necessary to include it?
Thanks a lot for your help,
Regards,
Ettore
logic proof-theory meta-math
edited Nov 29 '18 at 11:08
Mauro ALLEGRANZA
64.5k448112
asked Nov 29 '18 at 10:51
Ettore
969
"containing" : nothing misterious here : it means that $mathsf {PA}$ is made of the axioms of $text {EA}$ plus the induction axiom schema.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:08
The role of Chinese remainder th in the Godel encoding is dealed with in any math log textbook; see e.g. Mendelson's one. See Gödel numbering for sequences.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:10
PA: yes, but he uses 'containing' here in a wrong way. He just defined it to mean something else. That's very bad style, I think, but ok doesn't really matter..
– Ettore
Dec 1 '18 at 10:35
add a comment |
I am working through the above mentioned book, 'proof theory and logical complexity, volume 1', with some trouble here and there.
I would be glad if someone can help me with some of the exercises, clarify things when I can't work out the sense/meaning or help with the understanding of the proofs.
This is about example 1.4.6, I really have major problems here.
1) The author uses the word 'containing' in the first line. Apparently here he does not have in mind the definition of containment which comes RIGHT BEFORE this example, but instead he just means 'the axioms of EA and induction'. Right?
2) I don't see how to use theorem 1.3.5 to prove that PA contains PRA. (Here he uses the word 'contains' in accordance with his definition.)
3) I don't see what role is played by the chinese remainder theorem. I don't need to see a proof of it, and of course I can look it up online, but I don't see how it is used here, in this case, what exact formulation would be necessary etc.
4) What about the formula Prm(x,i,z)? What if x does NOT encode a sequence or i>=n. Then I guess the formula is supposed to be true, right?`
5) I have no idea what the long formula after 'The interpretation of $A_{Rfg}$ is' is saying and it is to me not at all 'immediate' that the interpretations of axioms (ix) and (x) hold.
6) When he tells why PA is prim. rec., in the end, why does he skip axiom (xv) of EA? Wouldn't it be necessary to include it?
Thanks a lot for your help,
Regards,
Ettore
logic proof-theory meta-math
edited Nov 29 '18 at 11:08
Mauro ALLEGRANZA
64.5k448112
asked Nov 29 '18 at 10:51
Ettore
969
I am working through the above mentioned book, 'proof theory and logical complexity, volume 1', with some trouble here and there.
I would be glad if someone can help me with some of the exercises, clarify things when I can't work out the sense/meaning or help with the understanding of the proofs.
This is about example 1.4.6, I really have major problems here.
1) The author uses the word 'containing' in the first line. Apparently here he does not have in mind the definition of containment which comes RIGHT BEFORE this example, but instead he just means 'the axioms of EA and induction'. Right?
2) I don't see how to use theorem 1.3.5 to prove that PA contains PRA. (Here he uses the word 'contains' in accordance with his definition.)
3) I don't see what role is played by the chinese remainder theorem. I don't need to see a proof of it, and of course I can look it up online, but I don't see how it is used here, in this case, what exact formulation would be necessary etc.
4) What about the formula Prm(x,i,z)? What if x does NOT encode a sequence or i>=n. Then I guess the formula is supposed to be true, right?`
5) I have no idea what the long formula after 'The interpretation of $A_{Rfg}$ is' is saying and it is to me not at all 'immediate' that the interpretations of axioms (ix) and (x) hold.
6) When he tells why PA is prim. rec., in the end, why does he skip axiom (xv) of EA? Wouldn't it be necessary to include it?
Thanks a lot for your help,
Regards,
Ettore
logic proof-theory meta-math
logic proof-theory meta-math
edited Nov 29 '18 at 11:08
Mauro ALLEGRANZA
64.5k448112
asked Nov 29 '18 at 10:51
Ettore
969
edited Nov 29 '18 at 11:08
Mauro ALLEGRANZA
64.5k448112
asked Nov 29 '18 at 10:51
Ettore
969
edited Nov 29 '18 at 11:08
Mauro ALLEGRANZA
64.5k448112
edited Nov 29 '18 at 11:08
Mauro ALLEGRANZA
64.5k448112
edited Nov 29 '18 at 11:08
Mauro ALLEGRANZA
64.5k448112
64.5k448112
asked Nov 29 '18 at 10:51
Ettore
969
asked Nov 29 '18 at 10:51
Ettore
969
asked Nov 29 '18 at 10:51
Ettore
969
969
"containing" : nothing misterious here : it means that $mathsf {PA}$ is made of the axioms of $text {EA}$ plus the induction axiom schema.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:08
The role of Chinese remainder th in the Godel encoding is dealed with in any math log textbook; see e.g. Mendelson's one. See Gödel numbering for sequences.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:10
PA: yes, but he uses 'containing' here in a wrong way. He just defined it to mean something else. That's very bad style, I think, but ok doesn't really matter..
– Ettore
Dec 1 '18 at 10:35
add a comment |
"containing" : nothing misterious here : it means that $mathsf {PA}$ is made of the axioms of $text {EA}$ plus the induction axiom schema.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:08
The role of Chinese remainder th in the Godel encoding is dealed with in any math log textbook; see e.g. Mendelson's one. See Gödel numbering for sequences.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:10
PA: yes, but he uses 'containing' here in a wrong way. He just defined it to mean something else. That's very bad style, I think, but ok doesn't really matter..
– Ettore
Dec 1 '18 at 10:35
"containing" : nothing misterious here : it means that $mathsf {PA}$ is made of the axioms of $text {EA}$ plus the induction axiom schema.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:08
"containing" : nothing misterious here : it means that $mathsf {PA}$ is made of the axioms of $text {EA}$ plus the induction axiom schema.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:08
The role of Chinese remainder th in the Godel encoding is dealed with in any math log textbook; see e.g. Mendelson's one. See Gödel numbering for sequences.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:10
The role of Chinese remainder th in the Godel encoding is dealed with in any math log textbook; see e.g. Mendelson's one. See Gödel numbering for sequences.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:10
PA: yes, but he uses 'containing' here in a wrong way. He just defined it to mean something else. That's very bad style, I think, but ok doesn't really matter..
– Ettore
Dec 1 '18 at 10:35
PA: yes, but he uses 'containing' here in a wrong way. He just defined it to mean something else. That's very bad style, I think, but ok doesn't really matter..
– Ettore
Dec 1 '18 at 10:35
add a comment |
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"containing" : nothing misterious here : it means that $mathsf {PA}$ is made of the axioms of $text {EA}$ plus the induction axiom schema.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:08
The role of Chinese remainder th in the Godel encoding is dealed with in any math log textbook; see e.g. Mendelson's one. See Gödel numbering for sequences.
– Mauro ALLEGRANZA
Nov 29 '18 at 11:10
PA: yes, but he uses 'containing' here in a wrong way. He just defined it to mean something else. That's very bad style, I think, but ok doesn't really matter..
– Ettore
Dec 1 '18 at 10:35