# Tag Info

### Why can we use induction when studying metamathematics?

This is not an uncommon confusion for students that are introduced to formal logic for the first time. It shows that you have a slightly wrong expectations about what metamathematics is for and what ...

### Is mathematics just a bunch of nested empty sets?

Warning: personal opinion ahead! It depends upon what the meaning of the word "is" is. One way to think about it - not necessarily historically correct - is the following: that the axioms of set ...
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### What exactly is an equation?

One approach: An equation is a predicate, $P(x)$, of the form $s(x) = t(x)$ where $x$ is a free variable (or vector of free variables) and $s(x), t(x)$ are terms -- expressions which evaluate to ...
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### What exactly is an equation?

Short answer touching on foundations and notation. Mathematicians use equations to tell their readers that two expressions (the things on either side of the $=$ sign) are actually two (different) ...
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### How are metalogic proofs valid?

Short answer: yes, it is essentially a chicken-and-egg problem, or perhaps a hermeneutical circle or a spiral. The common interpretation of the incompleteness theorems is that this circularity cannot ...
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### What exactly is an equation?

The universe of equations does not have uniform semantics. Equations can be tautologies.  1=1, x+x=2x, \tan x = \frac{\sin x}{\cos x}, \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^x = \mathrm{e}^x, \...
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### Is mathematics just a bunch of nested empty sets?

In addition to Noah's answer: Viewing mathematics as a "bunch of nested empty sets" is insofar simplifying as mathematics is more than the objects used in mathematics. Logical deduction is not a ...
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### If set theory only contains the notions of “set” and “is a member of” as primitives, how can an axiom of set theory refer to a “formula”?

In formalizations of set theory (e.g. ZFC), this subset axiom you've described in usually called the axiom schema of specification. Note the use of the word "schema" in its name: this is ...
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### Why can we use induction when studying metamathematics?

This is not at all intended to be an answer to your question. (I like Henning Makholm's answer above.) But I thought you might be interested to hear Thoralf Skolem's remarks on this issue, because ...

### How are metalogic proofs valid?

This is actually not a disagreement with Carl Mummert's answer, but another way of thinking about it. No, it isn't circular. But you don't prove what you might imagine you prove. All actual proofs in ...
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### Why do mathematicians generally write definitions in “declarative” rather than “imperative style?

I cannot speak for all mathematicians, but one key difference in computer science versus mathematics is that objects in computer science are typically viewed as mutable (unless they are declared '...
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### Consistency of ZFC and proof by contradiction

how would we know that it is the contradiction we were searching for and NOT an inconsistency of our axioms? We don't. But either way, the conclusion holds (since, if the axioms themselves are ...
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### How are metalogic proofs valid?

I think Richard Rast's answer is basically right, but I want to add a little to it. Ordinary logic, as practiced by mathematicians, studies a formalized copy of mathematics. It studies that copy ...
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### Is induction something we take on faith?

My viewpoint is the same as in Noah's first comment: for me, induction is part of the essence of what I mean when I talk about the natural numbers, so the thing I take on faith is not that induction ...
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### Math that does not have infinity

There are two schools of mathematics that reject infinity as it is presented in classical mathematics. Finitism. Finitism is a philosophy of mathematics that takes the Zermelo-Frankel axioms as the ...
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### Is mathematics just a bunch of nested empty sets?

Since this question is tagged with 'philosophy' I am surprised no one has mentioned Benacerraf's identification problem (you can read a summary here). In short, there are infinitely many ways to ...
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### What does it really mean for a model to be pointwise definable?

The first point is to distinguish between internal and external properties. This is exacerbated by the fact that we're looking specifically at $\mathsf{ZFC}$, which is "doing double duty" in ...
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### Consistency of ZFC and proof by contradiction

The short answer is that there is no simple way to distinguish an inconsistency that comes from a proof by contradiction from an inconsistency that comes from working with inconsistent axioms. In ...
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### Is mathematics just a bunch of nested empty sets?

It means that people have been trying to boil everything down to the least possible amount of data, rules, and symbols. Axiomatisation is a part of this process. But besides this, unless you really ...
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### In logic, is 'P = Q,' where P and Q are propositions, ill-conceived?

There is nothing ill-conceived about thinking of what is usually written as $P \Leftrightarrow Q$ or $P \leftrightarrow Q$ as an equation between the propositions denoted by $P$ and $Q$. However, it ...
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### Why do models of ZF which are not $\omega$-models have non-standard formulas whose length is "infinitely large natural numbers"?

For every natural $n$, $\phi_n$ is a sentence, where $\phi_0$ is $\forall x\,(x=x)$ and $\phi_{n+1}$ is $(\phi_n\land\phi_n)$. By recursion, there is a sentence in the theory that codes this claim and ...
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### Why is everything geometrical modeled on $\Bbb R$?

There is an axiomatic characterization of the reals: they are the unique complete, ordered, archimedic field. All of these properties are important for what we classically think of as geometry. The ...
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### If solutions to an expression are just another expression, why are they considered solutions?

You're right that figuring out a property of a solution or just re-writing the solution in another language may seem useless. But a lot of the great problems in mathematics arise from attempting to ...
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### Have we found a Turing Machine for which halting/non-halting is unprovable?

As announced in postings on Scott Aaronson's blog (The 8000th Busy Beaver number eludes ZF set theory and Three announcements), here are two explicit Turing machines whose halting/nonhalting behavior ...
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### Is it possible to derive the axiom of induction from a construction of the natural numbers?

Yes, it is possible to prove that a "natural numbers type structure" satisfies the induction axiom. The proof will be carried out in the same set theory that is used to form the natural number type ...
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### Help, Foundations of Mathematics: The Hilbert/Epsilon Operator, Bourbaki and Godel's Theorems

I won't try to explain how Bourbaki, Church, and Frege fit together, because they are not only giving different foundational systems but foundational systems for different parts of mathematics. For ...
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### Are Mathematicians Pluralists About Math?

I can't really speak for mathematicians in general, but from what I've observed I wouldn't say we ascribe to pluralism. Many mathematicians - those not directly involved in logic - don't even think ...
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### Are results of relative consistency metatheorems?

Yes. Consistency results are meta-theorems. They are not about implications of statements in the language, but rather about the properties of theories, which are objects of the meta-theory. Now, in ...
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Yes, they are metatheorems in the sense that they are theorems about formal systems. But notice there is a fuzzy line as to what to call a metatheorem. If $\phi$ is a sentence in the language of set ...