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1

A note on Łoś theorem without the Axiom of Choice

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Usuba T.

Bulletin of the Polish Academy of Sciences. Mathematics

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2024

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tom Vol. 72, no 1

17--44

EN

We study some topics around Łoś’s theorem without assuming the Axiom of Choice. We prove that Łoś’s fundamental theorem on ultraproducts is equivalent to a weak form that every ultrapower is elementarily equivalent to its source structure. On the other hand, it is consistent that there is a structure M and an ultrafilter U such that the ultrapower of M by U is elementarily equivalent to M, but the fundamental theorem for the ultrapower of M by U fails. We also show that weak fragments of the Axiom of Choice, such as the Countable Choice, do not follow from Łoś’s theorem, even assuming the existence of non-principal ultrafilters.

2

A Purely Algebraic Proof of the Fundamental Theorem of Algebra

100%

Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2016

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tom 8

7-23

PL

Proofs of the fundamental theorem of algebra can be divided upinto three groups according to the techniques involved: proofs that rely onreal or complex analysis, algebraic proofs, and topological proofs. Algebraicproofs make use of the fact that odd-degree real polynomials have real roots.This assumption, however, requires analytic methods, namely, the intermediatevalue theorem for real continuous functions. In this paper, we developthe idea of algebraic proof further towards a purely algebraic proof of theintermediate value theorem for real polynomials. In our proof, we neither usethe notion of continuous function nor refer to any theorem of real and complexanalysis. Instead, we apply techniques of modern algebra: we extend thefield of real numbers to the non-Archimedean field of hyperreals via an ultraproductconstruction and explore some relationships between the subringof limited hyperreals, its maximal ideal of infinitesimals, and real numbers.

3

A Purely Algebraic Proof of the Fundamental Theorem of Algebra

86%

Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia

|

2016

|

tom 8

7-23

EN

Proofs of the fundamental theorem of algebra can be divided upinto three groups according to the techniques involved: proofs that rely onreal or complex analysis, algebraic proofs, and topological proofs. Algebraicproofs make use of the fact that odd-degree real polynomials have real roots.This assumption, however, requires analytic methods, namely, the intermediatevalue theorem for real continuous functions. In this paper, we developthe idea of algebraic proof further towards a purely algebraic proof of theintermediate value theorem for real polynomials. In our proof, we neither usethe notion of continuous function nor refer to any theorem of real and complexanalysis. Instead, we apply techniques of modern algebra: we extend thefield of real numbers to the non-Archimedean field of hyperreals via an ultraproductconstruction and explore some relationships between the subringof limited hyperreals, its maximal ideal of infinitesimals, and real numbers.