Can you express a logic system like S5 using only a Gödel number?
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Since logic systems are just statements and/or axioms, can we formulate a logic system gödel numbering the system itself so that the system becomes nothing but a gödel number? For instance the modal system S5 would become just a Gödel number? Is it true that any logical statement must have a Gödel number and are there statements which don't have a Gödel number?
Thank you in advance
logic modal-logic
asked Aug 12 '11 at 2:19
Niklas Rosencrantz
55731435
add a comment |
up vote
2
down vote
favorite
Since logic systems are just statements and/or axioms, can we formulate a logic system gödel numbering the system itself so that the system becomes nothing but a gödel number? For instance the modal system S5 would become just a Gödel number? Is it true that any logical statement must have a Gödel number and are there statements which don't have a Gödel number?
Thank you in advance
logic modal-logic
asked Aug 12 '11 at 2:19
Niklas Rosencrantz
55731435
2
It's not true that a logical system is just statements and axioms. There are also rules of inference.
– Zhen Lin
Aug 12 '11 at 2:59
Yes and no. Aren't they all just a Gödel number whether it's a statement, axiom or rules of inference? If we can write rules of inference with Gödel numbers then this way anything is just a Gödel number? Can you present a counterexample of anything in logic which doesn't have a Gödel number?
– Niklas Rosencrantz
Aug 12 '11 at 3:25
2
@Niklas: How about an uncountable language, which cannot be encoded by cardinality issues?
– Asaf Karagila♦
Aug 12 '11 at 4:46
1
@Niklas R: Yes, you are completely right. Anything in $S_5$, for example the sentences and the derivations, can be assigned a natural number index in close analogy to the familiar Gödel numbering. But Gödel did his indexing with a definite purpose in mind, the Incompleteness Theorem. Any indexing for $S_5$ would, similarly, need to be done for a definite purpose.
– André Nicolas
Aug 12 '11 at 7:46
That's very interesting. Thank you for the insights.
– Niklas Rosencrantz
Aug 12 '11 at 18:16
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Since logic systems are just statements and/or axioms, can we formulate a logic system gödel numbering the system itself so that the system becomes nothing but a gödel number? For instance the modal system S5 would become just a Gödel number? Is it true that any logical statement must have a Gödel number and are there statements which don't have a Gödel number?
Thank you in advance
logic modal-logic
asked Aug 12 '11 at 2:19
Niklas Rosencrantz
55731435
Since logic systems are just statements and/or axioms, can we formulate a logic system gödel numbering the system itself so that the system becomes nothing but a gödel number? For instance the modal system S5 would become just a Gödel number? Is it true that any logical statement must have a Gödel number and are there statements which don't have a Gödel number?
Thank you in advance
logic modal-logic
logic modal-logic
asked Aug 12 '11 at 2:19
Niklas Rosencrantz
55731435
asked Aug 12 '11 at 2:19
Niklas Rosencrantz
55731435
asked Aug 12 '11 at 2:19
Niklas Rosencrantz
55731435
asked Aug 12 '11 at 2:19
Niklas Rosencrantz
55731435
asked Aug 12 '11 at 2:19
Niklas Rosencrantz
55731435
55731435
2
It's not true that a logical system is just statements and axioms. There are also rules of inference.
– Zhen Lin
Aug 12 '11 at 2:59
Yes and no. Aren't they all just a Gödel number whether it's a statement, axiom or rules of inference? If we can write rules of inference with Gödel numbers then this way anything is just a Gödel number? Can you present a counterexample of anything in logic which doesn't have a Gödel number?
– Niklas Rosencrantz
Aug 12 '11 at 3:25
2
@Niklas: How about an uncountable language, which cannot be encoded by cardinality issues?
– Asaf Karagila♦
Aug 12 '11 at 4:46
1
@Niklas R: Yes, you are completely right. Anything in $S_5$, for example the sentences and the derivations, can be assigned a natural number index in close analogy to the familiar Gödel numbering. But Gödel did his indexing with a definite purpose in mind, the Incompleteness Theorem. Any indexing for $S_5$ would, similarly, need to be done for a definite purpose.
– André Nicolas
Aug 12 '11 at 7:46
That's very interesting. Thank you for the insights.
– Niklas Rosencrantz
Aug 12 '11 at 18:16
add a comment |
2
It's not true that a logical system is just statements and axioms. There are also rules of inference.
– Zhen Lin
Aug 12 '11 at 2:59
Yes and no. Aren't they all just a Gödel number whether it's a statement, axiom or rules of inference? If we can write rules of inference with Gödel numbers then this way anything is just a Gödel number? Can you present a counterexample of anything in logic which doesn't have a Gödel number?
– Niklas Rosencrantz
Aug 12 '11 at 3:25
2
@Niklas: How about an uncountable language, which cannot be encoded by cardinality issues?
– Asaf Karagila♦
Aug 12 '11 at 4:46
1
@Niklas R: Yes, you are completely right. Anything in $S_5$, for example the sentences and the derivations, can be assigned a natural number index in close analogy to the familiar Gödel numbering. But Gödel did his indexing with a definite purpose in mind, the Incompleteness Theorem. Any indexing for $S_5$ would, similarly, need to be done for a definite purpose.
– André Nicolas
Aug 12 '11 at 7:46
That's very interesting. Thank you for the insights.
– Niklas Rosencrantz
Aug 12 '11 at 18:16
2
2
It's not true that a logical system is just statements and axioms. There are also rules of inference.
– Zhen Lin
Aug 12 '11 at 2:59
It's not true that a logical system is just statements and axioms. There are also rules of inference.
– Zhen Lin
Aug 12 '11 at 2:59
Yes and no. Aren't they all just a Gödel number whether it's a statement, axiom or rules of inference? If we can write rules of inference with Gödel numbers then this way anything is just a Gödel number? Can you present a counterexample of anything in logic which doesn't have a Gödel number?
– Niklas Rosencrantz
Aug 12 '11 at 3:25
Yes and no. Aren't they all just a Gödel number whether it's a statement, axiom or rules of inference? If we can write rules of inference with Gödel numbers then this way anything is just a Gödel number? Can you present a counterexample of anything in logic which doesn't have a Gödel number?
– Niklas Rosencrantz
Aug 12 '11 at 3:25
2
2
@Niklas: How about an uncountable language, which cannot be encoded by cardinality issues?
– Asaf Karagila♦
Aug 12 '11 at 4:46
@Niklas: How about an uncountable language, which cannot be encoded by cardinality issues?
– Asaf Karagila♦
Aug 12 '11 at 4:46
1
1
@Niklas R: Yes, you are completely right. Anything in $S_5$, for example the sentences and the derivations, can be assigned a natural number index in close analogy to the familiar Gödel numbering. But Gödel did his indexing with a definite purpose in mind, the Incompleteness Theorem. Any indexing for $S_5$ would, similarly, need to be done for a definite purpose.
– André Nicolas
Aug 12 '11 at 7:46
@Niklas R: Yes, you are completely right. Anything in $S_5$, for example the sentences and the derivations, can be assigned a natural number index in close analogy to the familiar Gödel numbering. But Gödel did his indexing with a definite purpose in mind, the Incompleteness Theorem. Any indexing for $S_5$ would, similarly, need to be done for a definite purpose.
– André Nicolas
Aug 12 '11 at 7:46
That's very interesting. Thank you for the insights.
– Niklas Rosencrantz
Aug 12 '11 at 18:16
That's very interesting. Thank you for the insights.
– Niklas Rosencrantz
Aug 12 '11 at 18:16
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
You can assign Gödel numbers to formulae in modal logic but it will not necessarily be interesting.
For one thing, you won't be able to refer to those numbers within the modal system so you don't get self-reference.
Also, $S_5$ is a decidable theory, whereas Gödel essentially used Gödel numbers to show that Peano arithmetic is not decidable.
answered Nov 18 at 6:28
Bjørn Kjos-Hanssen
1,876818
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
You can assign Gödel numbers to formulae in modal logic but it will not necessarily be interesting.
For one thing, you won't be able to refer to those numbers within the modal system so you don't get self-reference.
Also, $S_5$ is a decidable theory, whereas Gödel essentially used Gödel numbers to show that Peano arithmetic is not decidable.
answered Nov 18 at 6:28
Bjørn Kjos-Hanssen
1,876818
add a comment |
up vote
1
down vote
You can assign Gödel numbers to formulae in modal logic but it will not necessarily be interesting.
For one thing, you won't be able to refer to those numbers within the modal system so you don't get self-reference.
Also, $S_5$ is a decidable theory, whereas Gödel essentially used Gödel numbers to show that Peano arithmetic is not decidable.
answered Nov 18 at 6:28
Bjørn Kjos-Hanssen
1,876818
add a comment |
up vote
1
down vote
up vote
1
down vote
You can assign Gödel numbers to formulae in modal logic but it will not necessarily be interesting.
For one thing, you won't be able to refer to those numbers within the modal system so you don't get self-reference.
Also, $S_5$ is a decidable theory, whereas Gödel essentially used Gödel numbers to show that Peano arithmetic is not decidable.
answered Nov 18 at 6:28
Bjørn Kjos-Hanssen
1,876818
You can assign Gödel numbers to formulae in modal logic but it will not necessarily be interesting.
For one thing, you won't be able to refer to those numbers within the modal system so you don't get self-reference.
Also, $S_5$ is a decidable theory, whereas Gödel essentially used Gödel numbers to show that Peano arithmetic is not decidable.
answered Nov 18 at 6:28
Bjørn Kjos-Hanssen
1,876818
answered Nov 18 at 6:28
Bjørn Kjos-Hanssen
1,876818
answered Nov 18 at 6:28
Bjørn Kjos-Hanssen
1,876818
answered Nov 18 at 6:28
Bjørn Kjos-Hanssen
1,876818
1,876818
add a comment |
add a comment |
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2
It's not true that a logical system is just statements and axioms. There are also rules of inference.
– Zhen Lin
Aug 12 '11 at 2:59
Yes and no. Aren't they all just a Gödel number whether it's a statement, axiom or rules of inference? If we can write rules of inference with Gödel numbers then this way anything is just a Gödel number? Can you present a counterexample of anything in logic which doesn't have a Gödel number?
– Niklas Rosencrantz
Aug 12 '11 at 3:25
2
@Niklas: How about an uncountable language, which cannot be encoded by cardinality issues?
– Asaf Karagila♦
Aug 12 '11 at 4:46
1
@Niklas R: Yes, you are completely right. Anything in $S_5$, for example the sentences and the derivations, can be assigned a natural number index in close analogy to the familiar Gödel numbering. But Gödel did his indexing with a definite purpose in mind, the Incompleteness Theorem. Any indexing for $S_5$ would, similarly, need to be done for a definite purpose.
– André Nicolas
Aug 12 '11 at 7:46
That's very interesting. Thank you for the insights.
– Niklas Rosencrantz
Aug 12 '11 at 18:16