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Group Theory and Cardinals [duplicate]

I was wondering the following: Let's suppose that $G$ is a non-empty set, then, can we always find a binary operation $*$ such that $(G,*)$ is a group? For example, if we fix $G=\mathbb{Q}$ the sum ...
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Law of Excluded Middle Controversy

I was reading an introductory book on logic and it mentioned in passing that the Law of Excluded Middle is somewhat controversial. I looked into this and what I got was the intuistionists did not ...
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Creating mathematics vs building houses

I found the following quote in the book "Calculus" by Michael Spivak. (At the first page of Part 5,Epilogue, where he will discuss fields, construction of the real number, and uniqueness of the real ...
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Implication as defined in mathematical logic [duplicate]

If we can take the truth of an implication to mean the validity of reasoning, then that would mean that all reasoning that begins with false premises is valid reasoning. Are there no counterexamples ...
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An intuitive explanation for the negatives of divergent summations?

I am looking for an intuitive explanation for why divergent summations (that are always increasing) have finite values assigned as negative. An example that is beyond "because the math says so" kind ...
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1answer
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Prime Formulas in Heyting Arithmetic

I have been reading into Intuitionistic Logic, namely Heyting arithmetic, and I've bumped into this: Corollary 3.9. Let $A_0$ be a quantifier-free formula of $\mathscr L(HA)$. Then $$HA\vdash A_0\...
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Should it be allowed to apply classical logic to set theory?

It is well known, that the generalized continuum hypothesis isn't provable from the standard axiom system ZFC. GCH (generalized continuum hypothesis). For every infinite set A, there isn't a set M ...
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1answer
61 views

Evolution of Definitions

I try to understand how the definitions of mathematics have evolved (or formulated)... I'll use the epsilon-delta continuity definition as an example to ask my question... It may seem trivial, but it'...
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8answers
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What does it mean to solve an equation?

This question might be more philosophical than mathematical. In school we are taught how to solve equations such as $x^2 - 1 = 0$ or $\sin(x) - 1= 0$. Solutions to these equations are quite simple. ...
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How much of (pure) mathematics is first order logic?

Most automated theorem provers are built for first order logic only. How much are they missing out by focusing on first order logics?
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Did Ackermann produce a finitary consistency proof of second-order $PRA$?

In Wilhelm Ackermann's Doctoral Thesis (it is claimed, by Richard Zach, for one, in his paper "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program", arXiv: math/...
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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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Where do models of set theory live?

When we are studying independence proofs we are dealing with statements of the form $Cons(T)\rightarrow Cons(T')$ where $T$ and $T'$ are first order theories; commonly $T$ and $T'$ are subtheories of $...
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Does it make any sense to prove $0.999\ldots=1$?

I have read this post which contains many proofs of $0.999\ldots=1$. My question is, Does it make any sense to prove this equality? Can one give any "meaning" of the symbol $0.999\ldots$ ...
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What did Whitehead and Russell's “Principia Mathematica” achieve?

In philosophical contexts, the Principia Mathematica is sometimes held in high regard as a demonstration of a logical system. But what did Whitehead and Russell's Principia Mathematica achieve for ...
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Existence of mathematical objets constructed using the axiom of choice

Let consider the Vitali set $V \subset \mathbb R$, which is constructed using the axiom of choice. (I could take any other mathematical "object" that can be constructed using the axiom of choice, but ...
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1answer
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Representing a $\sigma$ - structure using a signature-$\sigma$ in Mathematical Logic.

In mathematical logic, I have a question regarding how a signature-$\sigma$ relates to a corresponding $\sigma$ structure which interprets the signature-$\sigma$ In Chiswell and Hodges book "...
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2answers
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The formal definition of an interval

I is A real interval iff ∀ x,y ∈ I the segment [x,y] ⊂ I I can't understand why an interval is defined this way Why it isn't defined the same way segments are? how can the definition of an object ...
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A box comprised of infinite number of small similar boxes.

On Wikipedia, I read, "A box can be thought of 'small boxes' infinitely repeating in all three dimensional directions" I don't understand what does Wikipedia wants to say with a box containing ...
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Are there alternative definitions/conceptions for the function concept?

The function is a mapping between elements of sets. However, I was thinking that would it be philosophically possible to come up with a new concept that's not a mapping between elements of sets, but ...
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Description of the Universe $V$ [closed]

For me, the concept "set" seams very ambiguous. This does not satisfy me because sets are used very often in mathematics, and so many questions in mathematics are not definite for me. I want to read ...
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5answers
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What is exactly a “Point”? [duplicate]

I read somewhere that a line is made up of infinite points. Between any two points on that line, there are another infinite points. and between any two points BETWEEN those 2 points there are ...
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“Probable” truth in mathematics

This might be more of a philosophical question, but why in mathematics is the tendency to only accept formal proof as a means of finding out what's true? In the physical sciences there's no such ...
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1answer
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Creating the formula and understanding them [closed]

We have, for example, the formula for velocity is distance over time ($\upsilon = s/t$). How do we get this formula? Why not ($t/s$ or $s*t$)? For this example we need physics to explain velocity, and ...
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2answers
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What “linguistic and philosophical problems” might be inherent in trigonometry?

In "A Mathematician’s Lament", Paul Lockhart derides the "status quo" of math education, claiming that "mathematics is an art form done by human beings for pleasure" but instead is taught "devoid of ...
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What does Poincaré mean for intuition of pure number?

To what does Poincaré refer in this article http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html speaking about the intuition of pure number? My answer is that he may refer to a ...
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3answers
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Math philosophy:about arithmetic operations and equality [closed]

This is some philosophic stuff about arithmetical operations that bothers me in some sense. During first grades, pupils are taught to perceive the arithmetical operations as "a process with a result". ...
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1answer
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Can there be a lottery of the natural numbers? [duplicate]

Can there be a lottery of the natural numbers, so that every natural number is chosen equally likely? The standard answer would be "No" because: If we define a measure $\mathbf{P}$ on $\mathbb{N}$ ...
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1answer
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What sort of sentence is the Goedel Sentence (for the First Incompleteness Theorem)?

As everyone knows who have studied Hilbert's writings, he divides sentences and terms (well-formed formulas) into at least (the 'at least' in deference to Smorynski) two classes: finitary, and ideal. ...
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1answer
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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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Value of a number

How does one define a value of a number? What is the value of the number 4? Asked differently, how does one show that a certain number is greater than another number? After this, one might ask how do ...
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3answers
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Is there an example for an undefinable number?

This question is motivated by a comment of Robert on the question Can any Real number be typed in a computer? : Can you "think of" an undefinable number? – Robert Israel I would like to ...
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If a property in $\mathbb{N}$ is true up to $10^{47}$ are there reasons to think it is probably true in all $\mathbb{N}$?

You have probably heard at some point statements like that the twin prime conjecture (namely that $2$ is an infinitely ocurring prime gap) is "probably" or "almost certainly" true. Same goes for a ...
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How different are the positive and negative numbers? [on hold]

Is there a fundamental difference between the positive and the negative numbers? Or is the difference like the one with electric charges in physics, where the other type of charge was just decided to ...
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552 views

Probability and measure theory

I'd like to have a correct general understanding of the importance of measure theory in probability theory. For now, it seems like mathematicians work with the notion of probability measure and prove ...
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1answer
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Logic (philosophy)

Let P(x), Q(x), and R(x) be open sentences containing the variable x. Prove that if the statements for all x (P(x) implies Q(x)) and for at least x (∃x) (Q(x) implies R(x)) are true, then for all x (P(...
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Intuitionistic response to Russel's Paradox

I'm having a look at intuitionistic approach to mathematics, and stumbled upon a derivation of Russell's Paradox that doesn't use the LEM. (Why did mathematicians take Russell's paradox seriously?)...
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Is there any solution to Frege's criticisms of Hilbert's Geometry without the application of Model Theory?

Recently I have come across the interesting debate of Frege and Hilbert regarding the Foundations of Geometry. It seems to me that the main concern of Frege was on the Logical Consistency of Hilbert's ...
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In logic why can't “p unless q” be “q -> ~p”?

Logically, when I think about p unless q I want to say that it is equivalent to q -> ~p, but the only equivalence is ...
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2answers
112 views

A philosophical question about an hypothetical theorem/equation of everything

Preamble I'm not a mathematician. I'm just curious. Please forgive my pseudo formalism. Please allow me, a non mathematician, to have just questions. Definition A mathematical theorem is a statement ...
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Can you choose -1 as the multiplicative unit? And what is a positive number?

If one starts with the cyclic group of integers and want to introduce multiplikation the ordinare choice of multiplicative identity is the generator 1. But since 1 and its inverse -1 is sort of the ...
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On the (Pre-)History of Sheaf Theory

In the wikipedia page on sheaf theory I found the following statement which somehow puzzled me: some of the facets of sheaf theory can also be traced back as far as Leibniz. Could anyone explain ...
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2answers
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Higher-Order Logic in ordinary Mathematics?

Do we use the language of higher-order logic in ordinary mathematics? (If yes: Can you give some examples?) Or are we always working with first-order logic? Comment: Maybe you are going to say that ...
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1answer
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Some Philosophical Questions About Mathematics and Logic [closed]

The following questions may seem very philosophical and I guess that you guys will tell me that this is not the right place for asking them. But for me it is important to get answers from ...
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Completeness property in signal analysis

Why completeness is an important property for signal analysis such as Fourier? What if we don't have a such a property?Many books discuss that the vector should not have a hole to complete.what is ...
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3answers
202 views

Logic & Reality [closed]

Maybe just a quick preface first before the question. I recently started a YouTube channel where I'm trying to clear up confusions I see on various (usually philosophical topics). In my 2nd video, the ...
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Question on a note by Kreisel

In Kreisel´s "Two notes on the foundation of set theory" he writes in a footnote that 2 is to be considered as measurable and omega is also measurable. Further more he goes on to say that an ...
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Meaning of “existence” for an uncomputable function related to the Halting Problem

Take the set of all Turing Machines $TM$, we can divide this set in two: $P$, the set of all Turing Machines that will halt if starting from an empty tape, and $Q$, its complement: the set of all ...
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What is the physical significance of arithmetic operations?

Here is an example of what I mean by physical significance: When we use some geometric or trigonometric identity, let us say Pythagoras' theorem to calculate the length of the diagonal of a field, ...
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Is it possible to distinguish rest and movement in hyperbolic universe?

Imagine a large body (for example, a planet) in 3D hyperbolic space. Now imagine the planet starts moving in a straight line at constant speed. In Euclidean space, all points would move along ...