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> Godel showed that there is no such thing as a universal logic in our current approach to formal systems, but that didn't suddenly make logic non-objective. It

> Godel showed that there is no such thing as a universal logic in our current approach to formal systems, but that didn't suddenly make logic non-objective. It simply means that there is no Ur-logic that can subsume all other logics (which is why most assert that Godel ended Hilbert's program). Could you elaborate on that? Any references?

naasking 4y ago
These are the implications of Godel's incompleteness theorems. No formal system expressive enough to encode arithmetic can simultaneously be both complete and prove its own consistency, because there will always be true propositions expressible in that system that cannot be proven in that system. This is why Hilbert's program to finitely axiomatize mathematics can't be completed. The "escape hatch" here is simply that not all propositions are actually interesting, so finite axiomatizations are still very useful, and we can extend the axiomatic basis as needed given satisfactory justification. This last part is the only place where social consensus sometimes comes into play (continuum hypothesis, etc). Edit: there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible...
- auggierose 4y ago
  Ah ok, I see what you mean. This is different from how I would understand the notion Ur-Logic: Just a logical system that can express anything you want, given you are free to add axioms. Obviously there are a few choices for that.
- qazxcvbnm 4y ago
  > there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible... Could you elaborate on what such a formalism where infinite structures are not expressible might look like? It sounds an intriguing notion, though I'm unable to quite get at it.
  - naasking 4y ago
    Bendegem's paper is a good introduction and overview: A Defense of Strict Finitism, http://www.jeanpaulvanbendegem.be/strict%20finitism.pdf http://www.jeanpaulvanbendegem.be/strict%20finitism.pdf

naasking 4y ago

These are the implications of Godel's incompleteness theorems. No formal system expressive enough to encode arithmetic can simultaneously be both complete and prove its own consistency, because there will always be true propositions expressible in that system that cannot be proven in that system. This is why Hilbert's program to finitely axiomatize mathematics can't be completed. The "escape hatch" here is simply that not all propositions are actually interesting, so finite axiomatizations are still very useful, and we can extend the axiomatic basis as needed given satisfactory justification. This last part is the only place where social consensus sometimes comes into play (continuum hypothesis, etc). Edit: there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible...

auggierose 4y ago
Ah ok, I see what you mean. This is different from how I would understand the notion Ur-Logic: Just a logical system that can express anything you want, given you are free to add axioms. Obviously there are a few choices for that.
qazxcvbnm 4y ago
> there is another possible escape hatch that hasn't been fully explored IMO, and that's some variant of finitism. All these impossibility proofs depend on infinite structures to derive incompleteness or contradiction, but if infinite structures are not expressible... Could you elaborate on what such a formalism where infinite structures are not expressible might look like? It sounds an intriguing notion, though I'm unable to quite get at it.
- naasking 4y ago
  Bendegem's paper is a good introduction and overview: A Defense of Strict Finitism, http://www.jeanpaulvanbendegem.be/strict%20finitism.pdf http://www.jeanpaulvanbendegem.be/strict%20finitism.pdf