# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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 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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### 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, ...
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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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### Is there an area of study regarding why certain mathematical definitions are useful?

Often in my studies I'll come across an definition, which I understand, and but don't necessarily see why the particular definition was chosen to be studied. For example, the topological axioms ...
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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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### How to show Universal Quantifier distributes over implication?

How to show Universal Quantifier distributes over implication? I've tried to no avail to show $\forall x(P(x) \implies Q(x))$ is equivalent to $\forall x(P(x)) \implies \forall (Q(x))$ but it seems no ...
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### What is the difference from a theorem and a meta-theorem?

I'm confused about what a meta-theorem exactly is and if a meta-theorem can be used to prove a theorem. To illustrate my confusion i give an example. Given the three statements: Every vector space ...
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### Meaning of symbols $\vdash$ and $\models$

I'm confused about the use of symbols $\vdash$ and $\models$. Reading the answers to Notation Question: What does $\vdash$ mean in logic? and What is the meaning of the double turnstile symbol ...
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### Good source on current 'views and thoughts' on mathematics

Recently I've become interested in the history of the way people think about mathematics. What I mean by this is for example how Godels proof basically put an end to the whole school of thought ...
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### Model-theoretic question about language of field theory.

Let $\mathscr{L}=\{+,·\}$ be the language of the theory of fields. Let $\phi$ be a sentence in this language. Show, using the compactness theorem of first-order logic, that if $\phi$ holds in finite ...
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### Most general mathematical framework

One can think of the same mathematical object in many different ways. For example take $\mathbb{R}$. One can think of this as (assume necessary hypotheses and so on) As a group. As a one ...
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### Where can I learn about Mathematical Philosophy?

This is a very vague question, but a question nonetheless. I am becoming increasingly more interested in what can be vaguely categorized as Mathematical Philosophy, or more specifically perhaps, ...
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### Has the Gödel sentence been explicitly produced?

I do not pretend to know much about mathematical logic. But my curiosity was piqued when I read Hofstadter's Gödel, Escher, Bach, which tries to explain the proof of Gödel's first incompleteness ...
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### Where can I find an ontology of algebraic structures?

A group is a monoid where every element admits an inverse, A ring is a monoid under multiplication that distributes over a commutative group A field is a ring whose non-zero elements form a group ...
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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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### Can a non-classical logic be used as a meta-logic to develop classical logic?

I have read much about non-classical logics such that paraconsistent logics , relevance logics , substructural logics , non-monotonic logic and so on. I think that the meta-logic logicians use to ...
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### General view of Theorems

I'm trying to see almost all theorems ( at least the non-existential ones ) as affirming that some formula ( mostly of first-order logic language ) is a logical consequence of other formulas. So, ...
Let $\mathcal{L}$ be a first-order language with quantifier $\forall$, connectives $\neg$ and $\rightarrow$, a two-place predicate $E$ and a one-place function symbol $f$. There are no other ...