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TylerH
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Are all true statements provable? Is there an axiom or theorem that says one way or the other?
If you're talking about formal mathematical statements (stated in some first-order language), then "true" depends on how you interpret the formal expression as a meaningful statement, and "provable" depends on what axioms you have to start with.TylerH said:Are all true statements provable? Is there an axiom or theorem that says one way or the other?
Goodstein's theorem is a true fact about the natural numbers, but it's not provable from PA. What do you mean when you say "true under the Peano axioms"?TylerH said:Can Goodstein's be said to be true under the Peano axioms if it can't be proven within them?
This is false. If something is true in every model of the Peano axioms, then it's provable from the Peano axioms. Godel's completeness theorem says essentially:micromass said:Yes, you should research something called sigma 1-completeness. It tells us that any sigma 1-statement (which, I think, includes Goodstein's theorem) which is unprovable is in fact true. So Goodstein's theorem is true (in the sense that every model of Peano's axioms satisfy Goodstein), but unprovable.
I'm not sure how it should be said. It could be restated as "true within a system governed by the Peano Axioms."AKG said:IfWhat do you mean when you say "true under the Peano axioms"?
AKG said:This is false. If something is true in every model of the Peano axioms, then it's provable from the Peano axioms. Godel's completeness theorem says essentially:
Fix a first order language [itex]\mathcal{L}[/itex]. If [itex]\phi[/itex] is an [itex]\mathcal{L}[/itex]-sentence, [itex]\Sigma[/itex] a set of [itex]\mathcal{L}[/itex]-sentences, then [itex]\phi[/itex] is true in every model of [itex]\Sigma[/itex] iff [itex]\Sigma[/itex] proves [itex]\phi[/itex].
In the context of Godel's incompleteness theorem, "true" means "true in the standard model [itex](\mathbb{N}, 0, S, +, \times, <)[/itex]."micromass said:A small question though: to my knowledge Godels incompleteness theorem tells us that there is a statement which is true, but not provable by Peano's axioms.
I always thought "true" was: it holds in every model. But this appears to be false. What do they mean with true then?
Godel's incompleteness theorem says, in some sense, PA is not strong enough to "govern" any system. There's a sentence [itex]\phi[/itex] such that there are two models [itex]M_1,\ M_2[/itex] such that both satisfy PA, but the first satisfies [itex]\phi[/itex] and the latter satisfies the negation [itex]\neg \phi[/itex]. So PA can't "govern" a system since it's not strong enough to tell systems it tries to govern how to behave, in particular it can't tell its systems whether to satisfy [itex]\phi[/itex] or its negation.TylerH said:I'm not sure how it should be said. It could be restated as "true within a system governed by the Peano Axioms."
AKG said:Godel's incompleteness theorem says, in some sense, PA is not strong enough to "govern" any system. There's a sentence [itex]\phi[/itex] such that there are two models [itex]M_1,\ M_2[/itex] such that both satisfy PA, but the first satisfies [itex]\phi[/itex] and the latter satisfies the negation [itex]\neg \phi[/itex]. So PA can't "govern" a system since it's not strong enough to tell systems it tries to govern how to behave, in particular it can't tell its systems whether to satisfy [itex]\phi[/itex] or its negation.
Normally, "true" means "true about the system of natural numbers [itex]\mathbb{N}[/itex]." In this case, there are true facts about [itex]\mathbb{N}[/itex] that PA can't prove.
There are structures such as:TylerH said:That helps, but the mention of PA was coincidental. In the general sense, does the fact a statement is true within some system imply it is provable within that system? Like with PA, for example, if PA was to be an axiomatic system, independent of other undeniable truths about N, would Goodstein's be true within that system?
You're welcome! Glad to help.TylerH said:Where's the like button? Seriously though, thanks. That explains it.
A true statement is a statement that accurately reflects or describes reality. It is a statement that can be objectively verified or proven to be correct through evidence or logical reasoning.
A statement that is always provable is more reliable and trustworthy. It ensures that the statement is not based on speculation or personal opinion, but on verifiable evidence or logical reasoning. This is crucial in scientific research and the pursuit of knowledge.
A statement can be considered always provable if it is supported by empirical evidence or if it can be logically derived from established principles or facts. It should also be able to withstand rigorous testing and scrutiny.
Yes, a statement can be true but unprovable. This can occur when there is insufficient evidence or knowledge to support the statement, or when the statement is beyond the scope of current scientific understanding. However, as scientific knowledge advances, these statements may become provable in the future.
No, not all scientific statements are always provable. Some statements may be untestable or unfalsifiable, meaning they cannot be proven or disproven through empirical evidence. However, these statements can still be useful in guiding further research and understanding of the natural world.