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Results tagged with metamathematics
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user 1946
the mathematical discipline that applies mathematical methods to the study of mathematical theories themselves.
7
votes
Accepted
How many variations can be derived from Gödel's fixed-point lemma?
Raymond Smullyan has several works, including several books, that explore diverse generalizations of the kind of fixed-points that your question is about.
Run the following search on MathSciNet: ti:(s …
- 236k
7
votes
Accepted
Can a general recursive function be defined by Pr(x)?
You haven't said which theory $T$ you intend to use, but probably you have in mind something like PA. You are proposing to define exponentiation $x^y$ essentially as "the smallest $z$ for which $T$ ca …
- 236k
4
votes
Accepted
Internal operations on uncomputable functions
The jump inversion theorem (Friedburg 1957) shows that any Turing degree $d$ above the halting problem is the jump of another degree $d=b'$, which means that $d$ is Turing equivalent to the halting pr …
- 236k
8
votes
Accepted
Can adding Hilbert's epsilon to a theory, and then expanding that theories axiom schema to i...
Even if you're looking at extensions of ZFC, the answer is still yes. Consider the theory of ZFC plus the scheme asserting that there is no definable global choice function. That is, the formulas "the …
- 236k
9
votes
Accepted
Can you formulate a theory stating that a truth predicate does not exist for first order set...
The assertion that there is (or is not) a truth predicate is expressible in the second-order language of set theory, but assuming consistency, not by any first-order assertion.
Second-order. In the …
- 236k
5
votes
Accepted
Is ZFC plus a truth predicate capable of variable substitution consistent?
The usual conception of truth predicate, or satisfaction class, is formulated in the way that you describe, allowing arguments. It doesn't handle only sentences (no free variables), but formulas, with …
- 236k
9
votes
Accepted
What two ordinals are these (based on definable ordinals)?
In this answer, let me assume as you indicated in the comments that you are working in a second-order set theory with a truth-predicate for first-order truth. Such a theory goes strictly beyond ZFC in …
- 236k
9
votes
What two ordinals are these (based on definable ordinals)?
There are several subtle issues with your post.
It is not in general possible to express the notion of "definable", because it leads to contradictions. For example, the class $D$ is not definable in …
- 236k
8
votes
On structures that are not submitted to compatibility conditions
When one studies a set with structures that do not interact with each other, then really all that matters about the set is its cardinality. And so the situation of your question can be viewed as arisi …
- 236k
12
votes
Accepted
Is there any set theory $T$ such that $T$ plus true arithmetic is complete with respect to s...
There can be no such theory $T$, even if you weaken the requirement
to $T$ being merely arithmetically definable, rather than insisting it must be effective.
To see this, consider the theory T+TA, wh …
- 236k
19
votes
Accepted
Is there a compendium of the consistency strength between the most important formal theories?
Cantor's Attic, which I founded with Victoria Gitman, is a compendium of information on the consistency strength hierarchy in set theory,
spanning the range from ZFC (which in this context is viewed a …
- 236k
23
votes
Accepted
Existential statement without witness
The answer is yes, provided these theories are consistent. For example, PA proves that there is a number $n$, such that if there is no proof of a contradiction in PA of size at most $n$, then there is …
- 236k
7
votes
Accepted
Rationale behind an requirement on Turing machines
Your proposed treatment of having machines use binary input with the alphabet $\{0,1\}$, where $0$ counts as a blank symbol (so that the input is padded with infinitely many additional $0$s, is not Tu …
- 236k
36
votes
Accepted
Meta$^{n{-}th}$ mathematics
Is the polynomial time
hierarchy regarded today as metamathematics? … This example therefore illustrates my point that there
is really no coherent distinction into
mathematics/metamathematics/meta-metamathematics. …
- 236k
8
votes
$\epsilon$-Formalization of Undecidability of CH
If ZFC is consistent, then no, it does not prove the assertion "CH is not provable in ZFC", since the non-provability of any assertion in a theory implies the consistency of that theory, and so this w …
- 236k