# Questions tagged [set-theory]

For questions about the mathematical branch that is based on the study of sets, i.e. collections of objects.

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### Where did Borel stress that $\mathbb{Q}$ being effectively enumerable by $\mathbb{N}$ is not about its size?

In the book 'Practical Foundations of Mathematics' (Paul Taylor), available online, one reads: As Emile Borel stressed in 1908, the important observation about $\mathbb{Q}$ [that there is a ...
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### Did Poincaré say that set theory is a disease?

This question has been discussed on several sites including MathOverflow but with not definite result. Presumely HSE is best suited. Jeremy Gray denies that Poincare said, "Later generations will ... 160 views

### Has Bolzano's opinion about the irrelevance of an infinite bijection ever found supporters?

From the only reason that two sets, $A$ and $B$, are corresponding to each other by the fact that for every part $a$ being in $A$ there is a part $b$ being in $B$ such that all pairs $(a + b)$ formed ...
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### What are the origins of Galileo's paradox?

Galileo's paradox is the observation that the natural numbers can be put into one to one correspondence with the square numbers, showing that an infinite set can be bijected to a proper subset of ...
71 views

### What were the inconsistent classes identified by Ernst Schröder?

From G.H.Moore’s Zermelo’s Axiom of Choice : In 1890 Schröder had already introduced a notion of consistent and inconsistent classes in his Algebra der Logik, well before the modern discovery of ...
416 views

### What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?

I read that Zermelo formulated AC in 1904 in order to formally prove the well-ordering theorem. Vitali’s 1905 proof of the existence of a non-measurable set of real numbers appears to the first “...
592 views

### How did Russell arrive at the paradox demonstrating the inconsistency of naive set theory?

As described here, we know that: In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted ...
403 views

### Did Bolzano conclude that $| \mathbb R | \ne | \mathbb N|$?

Boyer's History of Mathematics includes a brief account of Bolzano's contributions in the chapter on Gauss and Cauchy. Building on Galileo's "paradox" on the one-to-one correspondence between ...
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### History of the definition of Injective & Surjective Function

I'm a college student, just beginning to study Elementary Set Theory. In studying about the definition of injective and surjective function, out of curiosity, it came to my mind a question about the ...
699 views

### Did du Bois-Reymond invent the diagonal argument before Cantor?

The Wiki article on Cantor's diagonal argument mentions that the first use of a diagonal argument was in the work of Paul du Bois-Reymond in 1875. This would be one year after Cantor's first proof of ...
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### Kronecker vs Cantor — who won?

Now set theory is taught even to kids and it is the foundation of mathematics. Can we say that Cantor won?
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### History of $0 \in \mathbb N$.

There are different conventions around whether $0 \in \mathbb N$. I know that $\mathbb N = \{0,1,2,3, \cdots\}$ is called the Bourbakian definition of $\mathbb N$. But did this convention really ...
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### First use of curly braces to denote a set?

I was wondering who was the first person to Use curly braces to represent a finite set. Exempli gratia, $\{1,2,3\}$. Use set builder notation. Such as $\{2n:n \in \mathbb{Z}\}$ to represent the even ...
289 views

### Who was the first to mention the nine problem?

When Cantor published his seminal diagonal argument, he used binary code and did not distinguish the real numbers 0.1000… and 0.0111… Later, Koenig found a way to save the diagonal argument in binary ... 1k views

### What did Dedekind's The Nature and Meaning of Numbers contribute to the founding of Set Theory?

As best as I can tell Dedekind's paper was published in 1887 already several years after Cantor's flurry of papers on Set Theory between 1879-1883. With this in mind my central questions are: 1) What ...
930 views

### How much of mathematics did Russell's paradox really break?

According to my quite likely wildly oversimplified understanding, a revolution occurred in the foundations of mathematics when Cantor's formulation of set theory was found to be inconsistent due to ...
2k views

### How did mathematicians notate the empty set before $\varnothing$?

Recently, I learned that $\emptyset$ or $\varnothing$ is a relatively new notation for the empty set and was created in 1939. I know $\{\}$ is also used along with $\{\cdot\}$ to denote empty sets. ...
636 views

### Why did Cantor (and others) use $\mathfrak{c}$ for the continuum?

Kontinuum is German for continuum, but Cantor used $\mathfrak{c}$. Revision. J.W.Perry questions whether or not Cantor ever in fact used the symbol $\mathfrak{c}$. I must admit I just assumed that he ...
488 views

### Why is ZFC used more widely than NBG?

In studying the solutions proposed for Russell's Paradox on naive set theory (mainly the corresponding entry in the Stanford Encyclopedia of Philosophy) I came across the following considerations: von ...
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I do not remember the name/source of this paradox,but I remember I have discussed this with mathematicians and non mathematicians at least 5 times. It goes like this: "Every point of a line has ...
443 views

### Who discovered the difference between the infinities?

As we know, there is a difference between the (infinite) size (or cardinality) of the integer numbers and the size of the reals ($\aleph_0$ and $\mathfrak c=2^{\aleph_0}$). Who discovered it first?
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### Historical Instances of Set Theory

Context: I've been reading a lot about Set Theory lately, and how it suddenly sprung onto the mathematical scene in the late 1800's, thanks largely to Cantor. But it seems strange to me that no one ...
111 views

### Did Dedekind prove this lemma about posets (or an equivalent)?

Here's an easy lemma: Any poset $(S, \preceq)$ is order-isomorphic to a subset of the powerset $\mathcal{P}(S)$ ordered by set-inclusion. I seem to recall having seen this attributed to Dedekind. ...
752 views

### How much did scholasticism contribute to logic and mathematics?

I occasionally encounter mentions suggesting that late middle ages weren't as barren intellectually as commonly believed. For example, Occam and Scotus are credited with developing modal and ...
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### Did Galileo's writings on infinity influence Cantor?

To what extent was Cantor motivated by Galileo's paradox? More generally, to what extent were late 19th century mathematicians motivated by, or even aware of, Galileo's paradox? This is an issue I've ... 