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16 votes

Why did Hilbert believe consistency implies existence?

Hilbert wrote this observation some decades before the formal theories of completeness (and incompleteness) took shape, so you can't expect him to be overly precise with regard to such notions. The ...
Mikhail Katz's user avatar
  • 5,842
8 votes

Why did Hilbert believe consistency implies existence?

I just want to add some context to Katz's very nice answer. Hilbert's work on foundations occurred in the aftermath of the intuitionistic criticisms by Brouwer and Weyl. (Weyl in particular must have ...
Michael Weiss's user avatar
7 votes

Who was the first to write proofs in this manner?

Whitehead & Russel, Principia Mathematica (1910) is pretty much like this. The justifications are on the left, and only certain lines are numbered. But I doubt it is the first. Compare Frege's ...
Gerald Edgar's user avatar
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6 votes

When did mathematicians realize that theory of algebraically closed fields admits quantifier elimination?

A useful sketch of history is given in Alfred Tarski's Elimination Theory for Real Closed Fields by van den Dries. The result was known to Tarski by 1948 (when his Decision Method for Elementary ...
Conifold's user avatar
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4 votes

Claims that fully formal proofs are impossible to write down

In Donald Mackenzie's article, The Automation of Proof: A Historical and Sociological Exploration, I found some statements by P. Nidditch that come close to what I have been looking for. In the whole ...
Timothy Chow's user avatar
  • 1,516
4 votes

How could the people of the past be sure that $a \times b = b \times a$?

The guesses that are ventured in this question are entirely wrong. (1) That our reason to think multiplication is commutative is that we have an axiom that says so is wrong. (2) That those without ...
Michael Hardy's user avatar
4 votes

Why did Hilbert believe consistency implies existence?

Think back to when you learned linear algebra: \begin{align} x + y + z & = 2 \\ 6x - 4y + 5z & = 31 \\ 5x + 2y + 2z & = 13 \end{align} Does a solution to this system exist? (In this case, ...
Michael Hardy's user avatar
4 votes

Who first proved that empty set is subset of all sets?

Giuseppe Peano, Studi di Logica matematica (1897): Il segno $\Lambda$, fra classi, indica la classe nulla, cioè non contenente alcun individuo. Si può definire come segue [The $\Lambda$ symbols ...
Mauro ALLEGRANZA's user avatar
2 votes

Why did Hilbert believe consistency implies existence?

There is a historical context involved arising from the influence of Kant. There had been a number of reasons for anti-Kantian backlash in the nineteenth century. However, his wider influence led to ...
mls's user avatar
  • 29
1 vote

Why did Hilbert believe consistency implies existence?

@Dr. Katz The original poster concludes with the question: "Why does Hilbert believe that consistency of axioms for real numbers would provide him with existence of the reals?" So, the issue ...
mls's user avatar
  • 21

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