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lolgarithms
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what is the proof-theoretic strength (largest ordinal whose existence can be proved) for ZFC set theory?
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g_edgar said:If an ordinal [tex]\alpha[/tex] can be proved to exist, so can [tex]\alpha+1[/tex]
So in fact you probably want least ordinal whose existence cannot be proved
What? All I can do is make the obvious guess, and I'm not even sure that's well-defined, let alone the answer you seek.lolgarithms said:please, hurkyl, don't make me wait!
What? All I can do is make the obvious guess, and I'm not even sure that's well-defined, let alone the answer you seek.
The obvious guess would (IMHO) be something like the supremum of all of the ordinals in the constructible hierarchy.lolgarithms said:if the smallest ordinal that can't be proven is well-defined: what is the guess?
And that's where the extent of my knowledge ends. Although now that I think about it, the continuum hypotheses probably tells us some interesting information.CRGreathouse said:Perhaps it isn't well defined. What is known, then, about the supremum of the set of definable ordinals in ZFC, and the infimum of undefinable ordinals in ZFC?
The Proof-Theoretic Ordinal of ZFC is a mathematical concept that represents the strength of the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). It is the smallest ordinal number that cannot be proven to exist in ZFC. In other words, it is the highest level of induction that can be reached in ZFC.
The calculation of the Proof-Theoretic Ordinal of ZFC is a complex and ongoing process. It involves constructing a hierarchy of formal systems and determining the proof-theoretic ordinal of each system. Then, using a technique called the Feferman-Schütte ordinal notation, these ordinals are combined to calculate the Proof-Theoretic Ordinal of ZFC.
No, the Proof-Theoretic Ordinal of ZFC cannot be calculated exactly. This is because it is a limit ordinal, meaning that it is the highest possible ordinal that can be reached in ZFC. As such, its precise value is constantly changing and cannot be determined definitively.
Many other formal systems besides ZFC have a Proof-Theoretic Ordinal. Some notable examples include Peano Arithmetic, Gödel's System T, and Zermelo-Fraenkel set theory without the Axiom of Choice (ZF). Each of these systems has its own unique Proof-Theoretic Ordinal.
The Proof-Theoretic Ordinal is important in mathematics because it provides a way to compare the strength of different formal systems. It allows mathematicians to understand which systems are able to prove more statements and reach higher levels of induction. It also helps to classify different formal systems and study their properties.