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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.
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
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
3
votes
Deficiency of necessary conditions
First, let me mention that one must be careful when asserting that an implication fails. Taken literally, the assertion "D(x) does not imply P(x)" is logically equivalent to the assertion that D(x) i …
- 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
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
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
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
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
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
3
votes
Accepted
Given is "model". How many theories may it be a model?
You are using the terms "model" and "theory" in an idiosyncractic way.
In model theory, a model is a first-order structure, that is, a set with some functions, relations and perhaps distinguished el …
- 236k
3
votes
Accepted
Formal definition of 'useful' ?
It seems to me that you have two questions here.
First, you inquire about a formal account of "usefulness". I believe that this is already provided by the formal mathematical accounts of utility in …
- 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
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
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
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