# Tagged Questions

Meta-theory is the term for the theory in which mathematics is formalized (often PA, ZFC or similar theories). Meta-mathematical statements are statements which are evaluated at the level of the meta-theory rather than the theory. This tag is for questions regarding meta-mathematical theories, and ...

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### What would be the consequences if ZFC proved its own inconsistency, nonconstructively? [on hold]

Let's say a nonconstructive proof was given in ZFC that ZFC was inconsistent. Note that this doesn't automatically make ZFC inconsistent. Given a consistent theory $X$, $X + \neg \text{Con}(X)$ is ...
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### What is the best book on mathematical logic [on hold]

What is the best book on mathematical logic, the most complete, the most formal, and the most up to date? PS: price doesn't matter.
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### Could mathematical reasoning be non-axiomatic?

"Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, ...
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### What is the study of systems called?

Before I begin, i'll admit this isn't a maths question, however it is intrinsically a part of maths, and systems such as logic, set theory, and of course mathematics, will have been studied by many ...
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### Minimal arithmetic proving a statement similar to a Gödel sentence

I will use $\textbf{Q}$ to denote minimal arithmetic for this post. (I suppose Robinson arithmetic would also suffice (?)) Suppose we have $F(x)$ be a formula defining, in $\textbf{Q}$ the primitive ...
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### If a theory is 1-consistent then it is consistent

I am attempting to back two claims in this problem: I use $\textbf{Q}$ to denote minimal arithmetic for this post. I use the term 'rudimentary sentence' to denote formulas built using only negation, ...
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### The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
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### Weak Representability and Derivability Condition 1

Can someone point out the error in the following reasoning? Let K be an axiomatizable, consistent extension of Peano Arithmetic. Let P' denote the Gödel number for P. K is axiomatizable, thus Thm(k)...
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### Is there a standard name for using a function application, rather than a variable, as a summation index, as in $\sum_{f(x)}$?

I am trying to find out whether there is a standard notion of generalizing indexing such as $\sum_i$ to function applications as in $\sum_{f(x)}$. Intuitively, the latter means "iterating over all ...
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### Understanding order and countability

I am confused with how to defining the order of countable set. Let me express my thoughts by some examples: $\Bbb{N}$ has a 'natural' order, namely $1\lt2\lt3...$ For any countable set, we can define ...
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### Quantification over the set(?) of predicates

When learning set theory and logic, one fact that popped up a handful of times was that we did not quantify over predicates. To quote the notes I took in class on the axiom of unrestricted ...
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### Metamathematic: Cover the case if X=Y

I want to formalize: "If X is less than Y, Then U is equal to Y ", and have been told that $$\bf [\forall V \sim X=(Y+V)]U=Y$$ does not cover the case X=Y. Therefore I have rewritten it as  \bf [\...
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### Using the notion of provability only, how to show that $\Gamma \nvdash \varphi$?

For a practical example, suppose I want to show that $\{ P\} \nvdash Q$. From completeness, this is trivial: just find a model where $P$ is true and $Q$ false. But suppose I am stubborn and I don't ...
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### Is there a way of defining the notion of a variable mathematically?

I know that the notion of "set" is one that cannot be defined mathematically since it is the fundamental data type that is used to define everything else (and the definition which says that "sets" are ...
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### Primitive recursion and $\Delta^0_0$

Until recently I assumed that primitive recursive relations are exactly $\Delta^0_0$ (i.e. bounded) ones, but I learned they're different (the former is a proper superclass of the latter). I have ...
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### Why are all the interesting constants so small?

A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is ...
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### Is there a sense in which a formal theory can be more or less “metamathematically aware”?

For example, in classical logic, because of the law of the excluded middle, all propositions must be true or false, so in order to prove that a proposition is actually independent of the theory, we ...
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### Are there any coherent, complete, mathematical systems that do not imply the existence of the infinite?

Basically, are there any systems(complete with axioms) that do not imply the existence of an infinity? Is it possible to construct a mathematical system without infinity?
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### Is It True that We Can Never Be Sure of Validity of a Mathematical Proof?

The reason I ask this is because difficult mathematical proofs are just not plain self-evident. You would need a few years of intensive study before you can get to the point of understanding the ...
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### What does “dual statement” mean exactly in category theory?

I have long been confused about this notion. I know that for a statement within a single category, forming the dual statement is just reversing every arrows. But what about a statement concerning ...
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### Intuitonism and metamathematics.

There are various reasons why one would want to reject the law of the excluded middle when doing "normal" mathematics, which I won't get to here, but accepting those, does the same reasoning hold when ...
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### Why can we prove things via the contrapositive, contradiction, etc.?

I don't understand why we can use things as the contrapositive, reductio, etc. when we look to prove some statement. If we look at this from something like axiomatic propositional logic we can say ...
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### In meta-math does the sequence matter for $\bf ∀x$?

For example, if I have an axiom starting with $\bf ∀x∀x'\dots$ , would it still be an axiom if the only difference is that the sequence has changed to $\bf∀x'∀x\dots$?
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### When can independence of a statement in a theory be reduced to “truth”?

Since the Goldbach conjecture is in $\Pi_1^0$, if it were proven to be independent of Peano Arithmetic, it would follow that the Goldbach conjecture is true (i.e. true in the standard model), since ...
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### Formalizing the meta-language of First order Logic and studying it as a formal system

We've a formal system say First order Logic, we reason about it in our meta-language using our meta-logic. We study its properties as a mathematical object. We prove theorems like group theory. This ...
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### Discussion on AC and countable unions of countable sets

My question stems from the comments following Asaf Karagila's answer here : Why can't you pick socks using coin flips? At some point there was a discussion about whether or not a countable union ...
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### Artificial intelligence methods in mathematics

Are there aritficial intelligence methods in mathematics, automatic theorem discovery and proving? Google gives results in the opposite direction - mathematical methods of AI. Are there applications ...
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### Theorems relating to the limitation of mathematics

At one point, mathematicians believed that they may be capable of expressing all of mathematics in one system of ideas, and that their abilities were unlimited. Unfortunately, things like the Godel's ...
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### What did Hilbert actually want for his second problem?

When I read about the historical background of Gödel's incompleteness theorems, it is often mentioned that he was essentially responding to Hilbert, who was trying to prove the consistency of (...
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### Standard model of the Meta theory in the proof of arithmetic incompleteness

I have a question about the incompleteness of arithmetic. I wonder in which meta theory we are reasoning. For example, if we are in ZFC, then there are some non standard models of ZFC. As far as I ...
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### Needing an appropriate term for a metamathematical notion related with universes of discourse

I am working on a framework for metamathematics which uses structures of type (U, F, R), to which I refer as "universes" (short for "universe of discourse"), where U is a collection called "support" (...
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### [Paradox]How can Godel prove that Godel sentence is unprovable but true, if such proof itself proves that Godel sentence is true?

Isn't the proof that Godel sentence is unprovable but true a proof itself that Godel sentence is true? Godel in the preface of his proof remarked: “From the remark that [the unprovable statement] ...
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### For someone who loves mathematics - the more theoretical, the better - what type of engineering would you recommend?

I'm thinking there must be a type of engineering where math is strongly involved. What type of engineering would you suggest?
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### understanding the reducibility axiom

I am reading an english translation of Gödel's paper of the Incompleteness Theorem http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf In there is mentionned the so called "...
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### Order of Parentheses is Irrelevant: Metatheorem?

Here was shown by induction that the order of parentheses is irrelevant when associativity is verified. Question: Would this be a metatheorem about the formal language (say, of ZF) where the ...
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### Why in general there is no systematic way to find counterexamples? What kind of property do they all break that lead to this? and other things

We came across counterexamples in many areas of mathematics: For example Sum of irrational numbers not necessary being irrational The "Windmill blade" function (for lack of a better name of one of ...
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### Given any set of consistent axioms, is it always possible to find a model for these axioms in ZFC set theory?

If not, are there any conditions under which there must be a model under ZFC theory? Alternatively, is there any set of axioms for which this does hold true? If so, can we drop some of the axioms and ...
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### Is mathematics the only language that is not subject of interpretation?

Do you know any other "language" that is used by people except mathematics and is not subject of interpretation? By subject of interpratation I mean e.g. that 1 000 000 people will undertand that 1 + ...
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### Connection regularity in measure theory and approximation in premeasure

In the measure theory lecture, we defined a measure-theoretic content as follows: $\mu: \mathscr{C} \rightarrow [0,\infty]$ with the property being additive on disjoint sets and that the empty set ...
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### Help understand theorem that any set of first order sentences satisfied by N has a model that's a strict superset of N.

I saw the following theorem (in Computational Complexity book by Papadimitriou, p. 111) : Theorem : If $\Delta$ is a set of first-order sentences such that $N$ $\vDash \Delta$, then there ...
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### Separability in Algebra and Topology

I am wondering about the use of the word separablity in two different areas of mathematics, namely algebra and topology. In topology, we call a topological space $(X,\mathscr{T})$ if it contains a ...
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### Relationship between completeness and well ordering (meta).

Here is the definition for completeness of the reals (there are many equivalent formulations but I am interested in this one); Completeness: Every non-empty subset of the reals which is bounded above ...
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### Is this visual analogy to Gödel's incompleteness theorem accurate?

Today I was trying to explaining the Gödel's theorem to a layman, I've drawn a figure similar to the one below and said that: A truth is a consequence of the axioms (with the axioms also being truth)...
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### Are Euclid's Axioms Non-Logical or Logical

This question may seem trivial, but I recently became aware of the distinction between the two types of axioms: Logical and non-logical. What category does Euclid's fall under? I would assume they ...
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### Prove the Dual-Converse Relationship

Using a Hilbert-type system, given $E \sim F ,E' \sim F', E \implies F$ prove $F' \implies E'$ where a propositional formula A' is obtained by interchanging & and V in A. My answer is ...
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### Prove $A \supset B , B \supset C \vdash A \supset C$ [closed]

From Kleene's Introduction to Metamathematics, page 94 : "as further examples [of deduction] the reader may establish : " $A \supset B , B \supset C \vdash A \supset C$. (Here \$\...
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### Dependence of axioms of Zermelo set theory

I'm reading a book about Zermelo–Fraenkel set theory, and I was told that there are a lot of dependence relation among the commonly used axiom system (in the question What axioms does ZF have, exactly?...