2 added 187 characters in body
source | link

Any effectively axiomatized formal system that extends a very basic theory of formal arithmetic called Robinson Arithmetic (Q) will contain an undecidable sentence. In full generality, you can state the syntactic version of the First Incompleteness Theorem as follows:

(G1T) For any effectively axiomatized theory T that extends Q there exists a T-sentence G such that: (i) If T is consistent then T cannot prove G (ii) If T is omega-consistent then T cannot prove ¬G.

This is as general a statement of the theorem as you can get. Note that Kleene - in your quotation - is talking about the semantic version of the theorem. What Godel originally proved however involved no semantic notions such as truth, arithmetical truth etc. Whether or not you can see/recognize/acknowledge G as true depends on the semantic/intensional understanding of G - it is a common misconception that Godel wanted to prove the limitations of arithmetical truth. The background he was working against was the formalist delirium of the HilbertiansHilbertian formalism - so his primary concern was to demonstrate the purely syntactic limitations of formal systems. So, in full syntactic generality, (G1T) says (informally) that no approriate manipulation of symbols in any formal system complicated enough to contain some arithmetic will end up with G.

So you can see that (G1T) applies to a vast array of formal systems. Whether you choose to see the G's obtained as 'unprovable truths' depends on your semantics/ on whether your T has an intended interpretation and so on. But in most usual cases that kind of conclusion is considered justified. But it only arises because of the subvenient syntactic glitch that (G1T) ensures.

Now as for philosophical implications, the literature is immense. I will mention some pertinent issues:

  1. Some people have seen (semantic) (G1T) as a cogent argument against mechanism, i.e. against the thesis that minds are machines. The locus classicus is Lucas' Minds, Machines and Godel. Penrose extends and fortifies the argument in his books on consciousness. Ripostes to these arguments abound, see especially Putnam's Minds and Machines.
  2. Michael Dummett in the aptly titled 'The Philosophical Significance of Godel's Theorem' has argued that (G1T) may be construed as an argument against the thesis that meaning is use, by demonstrating to us that the use of any symbolic manipulation is always outrun by arithmetical truth and meaning. He introduces the notion of indefinite extensibility to salvage the thesis and provokes a lot of debate along thhe way.
  3. There are certain versions of Hilbert's Programme (HP) that can be said to have been refuted by (G1T). Nevertheless it is usually the Second Incompleteness Theorem that most people take to be the final nail in the coffin of (HP). Arguably this is the most monumental philosophical contribution of Godel's epoch-making discovery, namely that it single-handedly refuted Hilbertian formalism. Although, again, there are some research projects that can be seen as partial realizations of HP that are being carried out to this day - especially noteworthy in this direction are Detlefsen and Feferman.

Finally, I should like to say that countless philosophically-minded authors/thinkers/hacks have appropriated Godel's Theorem to try and make points about self-reference/loops/God/metaphysics/environmental awareness and God knows what else. The three implications I raised may seem dry and academic but they are also precise, well-studied and free from hyperbole. There is no doubt that Godel's theorems propelled our understanding of formal systems forward - and with it, our understanding of the philosophical powers and weaknesses of formal reasoning.

Any effectively axiomatized formal system that extends a very basic theory of formal arithmetic called Robinson Arithmetic (Q) will contain an undecidable sentence. In full generality, you can state the syntactic version of the First Incompleteness Theorem as follows:

(G1T) For any effectively axiomatized theory T that extends Q there exists a T-sentence G such that: (i) If T is consistent then T cannot prove G (ii) If T is omega-consistent then T cannot prove ¬G.

This is as general a statement of the theorem as you can get. Note that Kleene - in your quotation - is talking about the semantic version of the theorem. What Godel originally proved however involved no semantic notions such as truth, arithmetical truth etc. Whether or not you can see/recognize/acknowledge G as true depends on the semantic/intensional understanding of G - it is a common misconception that Godel wanted to prove the limitations of arithmetical truth. The background he was working against was the formalist delirium of the Hilbertians - so his primary concern was to demonstrate the purely syntactic limitations of formal systems. So, in full syntactic generality, (G1T) says (informally) that no approriate manipulation of symbols in any formal system complicated enough to contain some arithmetic will end with G.

So you can see that (G1T) applies to a vast array of formal systems. Whether you choose to see the G's obtained as 'unprovable truths' depends on your semantics/ on whether your T has an intended interpretation and so on. But in most usual cases that kind of conclusion is considered justified. But it only arises because of the subvenient syntactic glitch that (G1T) ensures.

Now as for philosophical implications, the literature is immense. I will mention some pertinent issues:

  1. Some people have seen (semantic) (G1T) as a cogent argument against mechanism, i.e. against the thesis that minds are machines. The locus classicus is Lucas' Minds, Machines and Godel. Penrose extends and fortifies the argument in his books on consciousness. Ripostes to these arguments abound, see especially Putnam's Minds and Machines.
  2. Michael Dummett in the aptly titled 'The Philosophical Significance of Godel's Theorem' has argued that (G1T) may be construed as an argument against the thesis that meaning is use, by demonstrating to us that the use of any symbolic manipulation is always outrun by arithmetical truth and meaning. He introduces the notion of indefinite extensibility to salvage the thesis and provokes a lot of debate along thhe way.
  3. There are certain versions of Hilbert's Programme (HP) that can be said to have been refuted by (G1T). Nevertheless it is usually the Second Incompleteness Theorem that most people take to be the final nail in the coffin of (HP). Arguably this is the most monumental philosophical contribution of Godel's epoch-making discovery, namely that it single-handedly refuted Hilbertian formalism.

Finally, I should like to say that countless philosophically-minded authors/thinkers/hacks have appropriated Godel's Theorem to try and make points about self-reference/loops/God/metaphysics/environmental awareness and God knows what else. The three implications I raised may seem dry and academic but they are also precise, well-studied and free from hyperbole. There is no doubt that Godel's theorems propelled our understanding of formal systems forward - and with it, our understanding of the philosophical powers and weaknesses of formal reasoning.

Any effectively axiomatized formal system that extends a very basic theory of formal arithmetic called Robinson Arithmetic (Q) will contain an undecidable sentence. In full generality, you can state the syntactic version of the First Incompleteness Theorem as follows:

(G1T) For any effectively axiomatized theory T that extends Q there exists a T-sentence G such that: (i) If T is consistent then T cannot prove G (ii) If T is omega-consistent then T cannot prove ¬G.

This is as general a statement of the theorem as you can get. Note that Kleene - in your quotation - is talking about the semantic version of the theorem. What Godel originally proved however involved no semantic notions such as truth, arithmetical truth etc. Whether or not you can see/recognize/acknowledge G as true depends on the semantic/intensional understanding of G - it is a common misconception that Godel wanted to prove the limitations of arithmetical truth. The background he was working against was Hilbertian formalism - so his primary concern was to demonstrate the purely syntactic limitations of formal systems. So, in full syntactic generality, (G1T) says (informally) that no approriate manipulation of symbols in any formal system complicated enough to contain some arithmetic will end up with G.

So you can see that (G1T) applies to a vast array of formal systems. Whether you choose to see the G's obtained as 'unprovable truths' depends on your semantics/ on whether your T has an intended interpretation and so on. But in most usual cases that kind of conclusion is considered justified. But it only arises because of the subvenient syntactic glitch that (G1T) ensures.

Now as for philosophical implications, the literature is immense. I will mention some pertinent issues:

  1. Some people have seen (semantic) (G1T) as a cogent argument against mechanism, i.e. against the thesis that minds are machines. The locus classicus is Lucas' Minds, Machines and Godel. Penrose extends and fortifies the argument in his books on consciousness. Ripostes to these arguments abound, see especially Putnam's Minds and Machines.
  2. Michael Dummett in the aptly titled 'The Philosophical Significance of Godel's Theorem' has argued that (G1T) may be construed as an argument against the thesis that meaning is use, by demonstrating to us that the use of any symbolic manipulation is always outrun by arithmetical truth and meaning. He introduces the notion of indefinite extensibility to salvage the thesis and provokes a lot of debate along thhe way.
  3. There are certain versions of Hilbert's Programme (HP) that can be said to have been refuted by (G1T). Nevertheless it is usually the Second Incompleteness Theorem that most people take to be the final nail in the coffin of (HP). Arguably this is the most monumental philosophical contribution of Godel's epoch-making discovery, namely that it single-handedly refuted Hilbertian formalism. Although, again, there are some research projects that can be seen as partial realizations of HP that are being carried out to this day - especially noteworthy in this direction are Detlefsen and Feferman.

Finally, I should like to say that countless philosophically-minded authors/thinkers/hacks have appropriated Godel's Theorem to try and make points about self-reference/loops/God/metaphysics/environmental awareness and God knows what else. The three implications I raised may seem dry and academic but they are also precise, well-studied and free from hyperbole. There is no doubt that Godel's theorems propelled our understanding of formal systems forward - and with it, our understanding of the philosophical powers and weaknesses of formal reasoning.

1
source | link

Any effectively axiomatized formal system that extends a very basic theory of formal arithmetic called Robinson Arithmetic (Q) will contain an undecidable sentence. In full generality, you can state the syntactic version of the First Incompleteness Theorem as follows:

(G1T) For any effectively axiomatized theory T that extends Q there exists a T-sentence G such that: (i) If T is consistent then T cannot prove G (ii) If T is omega-consistent then T cannot prove ¬G.

This is as general a statement of the theorem as you can get. Note that Kleene - in your quotation - is talking about the semantic version of the theorem. What Godel originally proved however involved no semantic notions such as truth, arithmetical truth etc. Whether or not you can see/recognize/acknowledge G as true depends on the semantic/intensional understanding of G - it is a common misconception that Godel wanted to prove the limitations of arithmetical truth. The background he was working against was the formalist delirium of the Hilbertians - so his primary concern was to demonstrate the purely syntactic limitations of formal systems. So, in full syntactic generality, (G1T) says (informally) that no approriate manipulation of symbols in any formal system complicated enough to contain some arithmetic will end with G.

So you can see that (G1T) applies to a vast array of formal systems. Whether you choose to see the G's obtained as 'unprovable truths' depends on your semantics/ on whether your T has an intended interpretation and so on. But in most usual cases that kind of conclusion is considered justified. But it only arises because of the subvenient syntactic glitch that (G1T) ensures.

Now as for philosophical implications, the literature is immense. I will mention some pertinent issues:

  1. Some people have seen (semantic) (G1T) as a cogent argument against mechanism, i.e. against the thesis that minds are machines. The locus classicus is Lucas' Minds, Machines and Godel. Penrose extends and fortifies the argument in his books on consciousness. Ripostes to these arguments abound, see especially Putnam's Minds and Machines.
  2. Michael Dummett in the aptly titled 'The Philosophical Significance of Godel's Theorem' has argued that (G1T) may be construed as an argument against the thesis that meaning is use, by demonstrating to us that the use of any symbolic manipulation is always outrun by arithmetical truth and meaning. He introduces the notion of indefinite extensibility to salvage the thesis and provokes a lot of debate along thhe way.
  3. There are certain versions of Hilbert's Programme (HP) that can be said to have been refuted by (G1T). Nevertheless it is usually the Second Incompleteness Theorem that most people take to be the final nail in the coffin of (HP). Arguably this is the most monumental philosophical contribution of Godel's epoch-making discovery, namely that it single-handedly refuted Hilbertian formalism.

Finally, I should like to say that countless philosophically-minded authors/thinkers/hacks have appropriated Godel's Theorem to try and make points about self-reference/loops/God/metaphysics/environmental awareness and God knows what else. The three implications I raised may seem dry and academic but they are also precise, well-studied and free from hyperbole. There is no doubt that Godel's theorems propelled our understanding of formal systems forward - and with it, our understanding of the philosophical powers and weaknesses of formal reasoning.