3 ms·
>The truth of a proposition P, within F, means "P is an axiom of F, or can be proved in F". No, truth and proof are NOT the same thing. The system F needn't ha
by xamuel 7y ago
>The truth of a proposition P, within F, means "P is an axiom of F, or can be proved in F".
No, truth and proof are NOT the same thing. The system F needn't have any notion of truth. For example, Peano arithmetic has no notion of truth. Peano arithmetic has notions of zero, successor, addition, and multiplication--none of truth.
Almost all competent mathematicians would agree that there exist theorems in the language of PA which are neither provable nor disprovable in PA but which are nevertheless true. Here, "true" means "true in the standard model of PA", not "true according to PA"---the latter isn't even meaningful because there is no such thing as "truth according to PA".
- strbean 7y agoOut of curiosity, could you give some examples of such theorems?
- xamuel 7y ago1. The original example would be Goedel's own example, namely, the statement (formulated in the language of PA) that PA is consistent, i.e., that PA does not prove 1=0. Less contrived examples are: 2. Goodstein's Theorem https://en.wikipedia.org/wiki/Goodstein%27s_theorem https://en.wikipedia.org/wiki/Goodstein%27s_theorem 3. A theorem stating that every computable strategy is a winning strategy in the hydra game (see https://faculty.baruch.cuny.edu/lkirby/accessible_independence_results.pdf https://faculty.baruch.cuny.edu/lkirby/accessible_independen...) 4. The strengthened finite Ramsey theorem, see https://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem https://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theor...
- strbean 7y agoThanks!