Natural language examples for failure of double negation elimination
I am trying to explain why double negation elimination $\neg \neg \phi \vdash \phi$ is invalid in intuitionistic logic, but introduction $\phi \vdash \neg \neg \phi$ is valid. The latter is easy: if I...
View ArticleIs there any logic system which ENTIRELY rejects non-contradiction of any...
I've recently learned about paraconsistent and intuitionistic logic, and dialetheism.According to the Stanford Encyclopedia of Philosophy's page on Dialetheism, it states:Dialetheism is the view that...
View ArticleMathematical induction, is it really a proof of truth?
This soft question is, I think, borderline mathematical philosophy.One of my former professors used to say that induction does not prove that a statement is true, but it merely eliminates the...
View ArticleWhat is the weakest set theory in which the set of all ordinals cannot exist?
The title may be confusing, I'll try to clarify first.For me, the weakest set theory in general is the extension of classical predicate logic with equality by the axiom of extensionality. Let's call it...
View ArticleInconsistency as a tool for proving sentences independent of $\mathsf{ZFC}$
Researchers in inconsistent/paraconsistent mathematics seek sub-classical logics that can be extended with the axiom of extensionality and naive comprehension to form a theory $T$ satisfying that:$T$...
View ArticleThe Direct Strategy for Proving Implications Involving Quantifiers
I want to prove a statement of the form ∀x ∈ X [ P(x) ⇒ Q(x) ], where P(x) and Q(x) are statements about x in the set X.The purpose of this post is to make sense of a strategy used to prove the above...
View ArticleIntrinsic and extrinsic properties of sets
Can a distinction between intrinsic and extrinsic properties of general sets a) be defined rigorously andb) be used fruitfully? (References?)An intrinsic property of a set $M$ is supposed to be about...
View ArticleHow do I convince someone that $1+1=2$ may not necessarily be true?
Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical...
View ArticleCorrect definition of Hochschild homology
In many expositions and discussions of Hochschild homology, it is stated that the classical definition via the cyclic bar complex (with underived tensor products) is incorrect (call the obtained...
View ArticleSignificance of étale / pro-étale cohomology in condensed mathematics
For my master's thesis I am studying Scholze + Clausen's theory of condensed mathematics.They begin with defining the category of sheaves on a particular site, the site in question being the pro-étale...
View ArticleCan Self-Verifying Theories define their own truth predicate?
All of the following information is from Dan Williard's Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection PrinciplesSelf-verifying theories are theories can prove their...
View ArticleAny connection between Eddington and Maclane?
I am reading The Nature of the Physical World by Sir A.S. Eddington.Written in 1928, AFAIK it predates any category theory or thinking, what with MacLane still two years away from graduating Yale at...
View ArticleIs there any "good" definition for what constitutes "applied mathematics"?
Is there any "good" definition for what constitutes "applied mathematics"?Wikipedia lists stuff such as statistics, optimization. However, e.g these have certainly "pure mathematical" aspects to them....
View ArticleWhy do people separate syntax and semantics in mathematical logic?
This is a rather vague or maybe philosophical question. Basically I want to have a deeper understanding on the motivation of syntax-semantics separation in mathematical logic, since it struck me when I...
View ArticleDo "truly" infinite proofs exist?
An assumption underlying this earlier question was the existence (and greater expressive strength) of infinite proofs in logics like $\mathcal{L}_{\omega_{1}^{CK}, \omega}$ (based on, for example, the...
View Article"You cannot engage in argument unless you rely on the principle of...
A very good SEoP article "Aristotle on Non-contradiction" by Paula Gottlieb makes two interesting claims, one after the other:Claim 1:"Anyone asking for a deductive argument for PNC [the principle of...
View Articlewhat is the definition of Mathematics ? [closed]
we all study mathematics , and all of us learn mathematical methods to solve problems , we learn how to prove , how to think mathematically but the question is, what is mathematics ? how can we define...
View ArticleIntuitive Randomness, Uniform PDFs, and Bertrand's Paradox
Suppose that somebody asks us the following question: Consider a straight line segment that goes from 0 to 1 (inclusive). Suppose that a point is chosen at random on this line segment. What is the...
View ArticleWhy can't the assumption of the existence of inaccessible cardinals prove...
It is known that ZFC + inaccessible cardinals implies the consistency of ZFC. Now, for the sake of argument let's equate ZFC with mathematics (this is a very strong and contentious assumption).Does it...
View ArticleErrors of Euler interpretation?
To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in...
View ArticleHow do mathematical objects relate to the real world? (a little philosophy)
I am just going to give an example of what I mean using Skolem's Paradox. I don't want to get into Skolem;s Paradox itself or its "resolution."Skolem's showed that in first-order formulations of ZFC,...
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Is this a correct universal property of prime decompositions?
Let $\mathcal F(\mathbb N)$ be the set of all finite subsets of $\mathbb N$ and let $\pi:\mathcal F(\mathbb N) \to \mathbb N$ be the product $\pi(A)=\prod_{x \in A}x$.Can the function $\omega:\mathcal...
View ArticleWhy do so many “obvious” mathematical statements turn out to be surprisingly...
In mathematics, it’s common to encounter statements that feel intuitively obvious but turn out to be extremely subtle, difficult to prove, or even false upon closer inspection. Examples include:The...
View ArticleWhat makes the set membership a predicate (or, non-set)?
This excerpt is from Open Logic Project's Set Theory: An Open Introduction (p. 30):Third: when we “identify” relations with sets, we said that we wouldallow ourselves to write $Rxy$ for $\langle x,...
View ArticleDoes logic work this way [closed]
Let me describe a hypothetical but thought-provoking scenario:Two mathematicians, Person A and Person B, both prove the same mathematical result, let’s say X (for example, a weaker version of...
View ArticleDoes the epsilon-delta definition of limits truly capture our intuitive...
I've been delving into the concept of limits and the Epsilon-Delta definition. The most basic definition, as I understand it, states that for every real number $\epsilon \gt 0$, there exists a real...
View ArticleWhy is writing axioms informally considered part of naive set theory?...
Just to be clear I assume informal language means written in 'natural language' or the language I am using to write this question and formal notation means written in contemporary standard mathematical...
View ArticleAre declarative sentences with unknown truth value seen as statement [duplicate]
In logic,A statement is a declarative sentence that is either true or false, but not both.Now, let's think the following sentences:On November $23, 1894$, around $2000-5000$ black sheep were born in...
View ArticleWhat are the fundamental assumptions that lead to one extracting the correct...
This is a Distance/rate problem. The only info given is the formula: Distance = Rate * Time."Two cars are 500 miles apart and moving directly towards each other. One car is moving ata speed of 100 mph...
View ArticleReference books for Foundations of Mathematics (Logic and Philosophy)
I am an undergraduate student. And I want to build solid foundation for Mathematics.I tried google search but could not get proper recommendation.Please suggest books which covers the subject...
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