Who has actually read Godel's theorems?

[mathwonk]

in my admittedly untutored opinion, you still seem to be missing the point that there are statements which cannot be added as false to some systems of axioms, preserving consistency, and these are the interesting ones for incompleteness.

i.e. the negation of a true but unprovable statement cannot be added preserving conssitency. a statement is independent iff either it or its negation can be added both preserving consistency.

for example according to your reference, goodstein's theorem is true but undecidable in 1st order Peano arithmetic, i.e. the negation of goodsteins theorem cannot be added without sacrificing consistency.

there is no comparison between this situation and that of euclids parallel postulate, since either it or its negation can be added to the other postulates without sacrificing consistency.

there is nothing surprizing about a statement being independent of another collection of statements, except for the historical accident that it took people hundreds of years to notice the obvious, i.e. that "table top geometry" is a model for the rest of euclids system of axioms in which the 5th postulate is false.

the whole point of incompleteness is the existence of an unprovable statement whose negation cannot be added with sacrificng consistency.

of course i know, as with the parallel postulate, there are models of set theory in which either the continuum hypothesis or its negation are true. this is an example of independence, not incompleteness.

what am i missing?

 

[chronon]

for example according to your reference, goodstein's theorem is true but undecidable in 1st order Peano arithmetic, i.e. the negation of goodsteins theorem cannot be added without sacrificing consistency.

No, the negation of Goodstein's theorem can be added to 1st order Peano arithmetic without sacrificing consistency. In fact this is how undecidability is proved, a model of the axioms is constructed in which the theorem is false. (We already have a model in which it is true, the normal integers).

 

[HallsofIvy]

What exactly do you take as the meaning of "true but unprovable"? In other words, exactly what do you mean by "true" in an abstract axiom system?

 

[master_coda]

in my admittedly untutored opinion, you still seem to be missing the point that there are statements which cannot be added as false to some systems of axioms, preserving consistency, and these are the interesting ones for incompleteness.

This is not true if you are working with first-order logic. If you're working with first-order Peano arithmetic, a statement can be undecidable if and only if there exists models in which the statement is false and models in which the statement is true.

the whole point of incompleteness is the existence of an unprovable statement whose negation cannot be added with sacrificng consistency.

It is a consequence of incompleteness that higher-order logics have statements of this form. However the theorem itself is concerned with first order logic, so this is certainly not the "whole point" of incompleteness.

 

[jennycraig10]

I'm a high school student and am doing a report on Kurt Godel... Can anyone help me out by telling me what career there is out there that uses Godel's theory? I would really appreciate it.

 

[honestrosewater]

I'm a high school student and am doing a report on Kurt Godel... Can anyone help me out by telling me what career there is out there that uses Godel's theory? I would really appreciate it.

Teaching or doing research in logic, math, computer science or, less likely, philosophy. Minor point: it's theorem; see here.