# Tagged Questions

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### Need some help with this Cardinality/sets question.

I've got this problem about sets, and cardinality. I don't really understand it other than cardinality is the number of elements within each set, I don't understand a lot of the signs used within the ...
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### Countable Set & Formal Grammar

We know set A is countable if A is finite or in a one-to-one mapping to natural numbers. I try to summarize my though. I think the following proposition is true. suppose $\Sigma$ is arbitrary ...
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Axiom of specification is schema because it talks about definite condition(or wff) which use notion of finite but this again we define from sets. But in logic we defined wff using consept of tuple and ...
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### Showing that $A \cap X = A$ for all $A$ if and only if $X = S$.

I have the following task: Let $S$ be a nonempty set. All capital letters will denote subsets of $S$. Show that $A \cap X = A$ for all $A$ if and only if $X = S$. This does not seem to true. ...
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### How to prove that $max(\aleph_{0}, card(X)) = max(\aleph_{0}, card(L(X)))$?

I struggle with the following problem. Let $X$ be a set of elementary sentences and $L(X)$ be the smallest elementary language in which we can express all the sentences from $X$. How to prove that ...
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### Cantor's diagonal argument: the coherence of the sequence constructed to prove that 2[N] has a higher cardinality than N

I think I have a good understanding of how Cantor's argument for showing larger cardinalities works; but I do have a continuing problem with one part of it, namely the construction of the element ...
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### Does intuitionist logic deny diagonal argument?

Let us for example give an example of diagonal proof of uncountability of the set of real numbers $\mathbb{R}$. Would intuitionists accept this, or deny this? If they deny this argument, why would ...
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### Extensional versus intensional theories in mathematics

Lambda calculus is often cited as an intensional theory whereas set theory is cited as an extensional theory. What are other examples of extensional and intensional theories of mathematical logic?
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### Semantics and Logical structure in Definitons

Continuation of Free and bound variables in "if" statements definitions: A number is even if it is divisible by $2$. The number is even if it is divisible by $2$. Is the usage of the ...
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### Logical structure of definitions

Here are some "concepts" that are confusing me: The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion ...
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### Preimage simple problem

Which one is correct and which one is wrong ? $f^{-1}[Y \cap Y^{'}] \subseteq f^{-1}[Y] \cap f^{-1}[Y^{'}]$ $f^{-1}[Y] \cap f^{-1}[Y^{'}] \subseteq f^{-1}[Y \cap Y^{'}]$ Here is my solution: ...
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I have a quick question in regard to sets. I am a little confused when I see the notation $A\subseteq B$. How is this different than the sets $A$ and $B$ being identical? I guess some of the confusion ...
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### Simple fact about class of sets.

I have a simple question which is very trivial for the other people, I guess. However, I never fully understand the argument completely. Take any $X\neq\emptyset$ and let $\mathbf{S}$ be an arbitrary ...
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### Truth set of a Universal Quantifier and Family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A(A\in F \implies x\in A)\bigr\}$$ If my understanding serves me correctly this notation means that all the $x$ ...
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### Definition of intersection of a family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A: (A\in F \implies x\in A)\bigr\}$$ If my understanding serves me correctly this notation means that all the $x$ ...
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### The set of all philosophers is a camel

In Wilder's book "Introduction to the Foundations of Mathematics," a "proof" for the statement: "The set of all philosophers is a camel" is mentioned in passing as something done by Grelling under the ...
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### Formalizing the Fallacy of Composition

Is there a well-known formalization of the fallacy of composition? More generally, where in mathematics is it true that if a property holds for all of some elements of a set it holds for the whole ...
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### How to show that $\bigcup_{i\in \emptyset} A_i=\emptyset$ and $\bigcap_{i\in \emptyset}A_i=X$? [duplicate]

It was probably asked multiple times before which is about basic logic. But I couldnt find any. let X a set and I is index set. for $i\in I$, $A_i$ is a subset of X. if $I=\emptyset$, how can we show ...
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### A proof in naive set theory.

I am trying to use naive set theory to figure out a proof of the following statement: $$(x = u \land y = v) \to 〈x, y〉 = 〈u, v〉$$. What propositions should i use to prove this?
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### A* finite or infinite? (Set theory)

I have a question regarding the following: If A is a set, then by A* we mean the set of all finite rows of elements of A. Now suppose A is finite. How big is A*, and how can you see that? I ...
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### Prove that $A\subseteq B\Longleftrightarrow A\cap B = A$

In set theory logic mathematics. How would i do the proof for: $A\subseteq B\Longleftrightarrow A\cap B = A$
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### Convincing proofs, proofs by contradiction and countability

Disclaimer: I have a (modest) background in mathematical physics, not logic, so I know very little of the latter. Although when I understood Cantor's argument for the first time (from one of Martin ...
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### Countable and Uncountable sets

Is $\mathbb{N}\cup\{a\}$, for some $a\not\in\mathbb{N}$ countable or uncountable? $\mathbf{Attempt: }$ It is true that a set is countable if there exists an injective function $f : S → N$ from $S$ to ...
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### How did I solve this problem?

While writing a SQL query I had to solve a problem I'd never dealt with before. It was trivial, but I cannot explain the solution without drawing lines on paper or making examples with actual numbers ...
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### What's the difference between relations, functions, and operations?

My understanding is vague: operations is a subclass of functions, functions is a subclass of relations. operations and functions can form terms but not formulas, relations can form formulas but not ...
In some papers on mathematical logic I've found references to hierarchy like $\Sigma_1^0$-sentence and so on. What does it mean? What is $\Sigma_n^m$, what is $n$ and $m$ here?
I just want to make sure I'm negating the following logical statement correctly (for a contradiction proof): For every set $A$, there exists a well ordered set $V$ such that there exists no ...