Abstract
We start by explaining which objects are studied in mathematics. We follow the standard approach in which these objects are mathematical structures. We discuss the crucial role of sets in the foundations of mathematics and briefly sketch how they are used to define some basic arithmetical structures. In this introductory chapter we state the basic axioms about sets informally as principles. We present the most important antinomies of intuitive set theory. We discuss the axiomatic method, which was discovered in antiquity and is the basis of all contemporary mathematical theories. Finally, we explain why abstract concepts are useful, even if we only need to solve a problem stated in elementary terms.
The real universe arched sickeningly away beneath them. Various pretend ones flitted silently by, like mountain goats. Primal light exploded, splattering space-time as with gobbets of junket. Time blossomed, matter shrank away. The highest prime number coalesced quietly in a corner and hid itself away for ever.
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- 1.
‘Arity’ is not an English word, but it is common in mathematical jargon. The word is derived from the suffix -ary.
- 2.
Not to be confused with groupoids in category theory.
- 3.
The very recent result of T. Tao [289] that every odd integer greater than 1 can be represented as a sum of 5 or fewer primes uses the fact that the Goldbach conjecture has been verified by computation for all numbers up to 4⋅1014.
- 4.
In order to get full information about G 1, we need the groups Ker(f) and Im(f) and, furthermore, a homomorphism from Im(f) into the group of automorphisms of Ker(f). It would take us too far afield to explain this relation.
- 5.
I assume that we agree on what red things are.
- 6.
It is consistent to assume the existence of some sets x which are equal to {x}, but usually they are prohibited by other axioms, as they are rather unnatural.
- 7.
This media file is in the public domain in the United States.
- 8.
More precisely, we must also use definitions of arithmetical operations and axioms about sets.
- 9.
Ur-, originally a German prefix now also used in English, means primitive, original.
- 10.
From falsehood, it follows anything you like.
- 11.
Strictly speaking, we should distinguish between paradoxes—apparent contradictions, and antinomies—actual contradictions, but when using informal reasoning it is difficult to make this distinction. Therefore, these words are used interchangeably.
- 12.
Letters to Hilbert, September 26 and October 2, 1897. See [65], page 42.
- 13.
This is the traditional version of the paradox which assumes that a liar is always lying.
- 14.
[254], Vol. 1, page 36.
- 15.
Grundlagen der Geometrie, [124].
- 16.
Sometimes it is useful to keep some redundancy; sometimes we are not able to prove that further reduction is impossible, but it is.
- 17.
This is not quite precise. One has to first generalize the theory and only then it is possible to add Einstein’s Field Equations. The generalizations without the Field Equations are also interesting theories and can describe nontrivial phenomena.
- 18.
For this argument one can also use the simpler version in which only one square is cut out. The reason to cut out two is simply to make the puzzle a little harder. With only one square away the idea of counting the parity of the number of squares comes to one’s mind immediately.
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Main Points of the Chapter
Main Points of the Chapter
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A structure is given by a set of elements (the universe) and relations and operations (defined on the universe). There are infinitely many types of structures.
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Mathematicians study some standard structures, such as the natural numbers and the real numbers, and various classes of structures, such as groups. The basic structures were introduced a long time ago; others were defined more recently.
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Two basic principles for sets are extensionality and comprehension.
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Infinitely many sets can be constructed starting with a single set, the empty set.
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Relations and operations can be defined as sets of pairs, triples, etc. Therefore, we only need sets to formalize mathematical structures.
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Russell’s paradox destroys the hope of having a consistent set theory based solely on our intuition. We have to use axiomatic set theory.
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In mathematics, theories are defined by axioms. To derive theorems in an elementary theory we only need logic. However, in order to be able to state and prove interesting results, logic alone does not suffice; we have to use set theory.
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There are several reasons for using the axiomatic method: 1. it is precise, 2. it is fair because we state the assumptions explicitly, 3. it is useful because we can test whether the theory can be applied to a particular phenomenon, 4. it helps us to explain the studied concepts because a short list of basic axioms explains the essence better than a long complicated description.
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There are mathematical problems that can be stated in a completely elementary way, but cannot be solved without applying very abstract concepts.
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Mathematical logic has the means to measure the degree of abstractness of concepts and to prove that such concepts are indispensable. Until now, however, we have succeeded in proving the necessity of using abstract concepts only in a few, rather simple instances.
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Pudlák, P. (2013). Mathematician’s World. In: Logical Foundations of Mathematics and Computational Complexity. Springer Monographs in Mathematics. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00119-7_1
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