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2024 | OriginalPaper | Chapter

2. Embeddings and Substructures

Author : David Marker

Published in: An Invitation to Mathematical Logic

Publisher: Springer Nature Switzerland

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Abstract

We study embeddings of structures and how the truth of formulas is preserved under embeddings. Elementary embeddings are introduced and the Tarski–Vaught test for elementary embeddings and the Downward Löwenheim–Skolem Theorem are proved.

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previous chapter Languages, Structures, and Theories

next chapter Formal Proofs

There is an automorphism of \(F(t)\) fixing F and sending t to \(t+i\). Also, if K is a subfield of \({\mathbb C}\), then any automorphism of K extends to an automorphism of \({\mathbb C}\).

A somewhat surprising consequence occurs when studying models of set theory. Suppose \(\mathcal {M}=(M,\in )\) is a model of set theory, say \(\mathcal {M}\models \)ZF or ZFC and \(\in \) is the membership relation on \(\mathcal {M}\). There is \(\mathcal {N}\preceq \mathcal {M}\) with \(\mathcal {N}\) countable. But \(\mathcal {N}\models \exists x \)“x is uncountable.” We now need a bit of set theory. Taking the Mostowski collapse (see, for example, [57] III 5.9) we can find a transitive \(\mathcal {N}^\prime \cong \mathcal {N}\). Every element of \(\mathcal {N}^\prime \) is actually a countable set. Thus there is a countable element that \(\mathcal {N}^\prime \) believes is uncountable. This is called Skolem’s Paradox and has been extensively discussed by philosophers [79]. Of course, there is no real paradox. Suppose \(a\in \mathcal {N}^\prime \) and \(\mathcal {N}^\prime \models a\) is uncountable. This just means that no bijection between a and \({\mathbb N}\) is in \(\mathcal {N}^\prime \).

57.

Kunen, K.: Set Theory. An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, 102. North-Holland Publishing, Amsterdam (1983)

79.

Putnam, H.: Models and reality. J. Symbolic Logic 45(3), 464–482 (1980)MathSciNetCrossRef

Title: Embeddings and Substructures
Author: David Marker
Publisher: Springer Nature Switzerland
Book: An Invitation to Mathematical Logic
Print ISBN: 978-3-031-55367-7

Electronic ISBN: 978-3-031-55368-4

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-55368-4_2

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