Constant expansion

The constant extension or constant expansion is an important process in mathematical logic . A language is extended (expanded) by new constants in order to obtain certain desired properties in the extension. Then what is achieved in the expanded language is reduced back to the source language.

definition

The signature of a language, for example the first or second level predicate logic , contains, among other things, a possibly empty set of constant symbols. The expansion to a larger set , where and are disjoint, expands the source language with new constant symbols . ${\displaystyle \sigma }$ ${\displaystyle C_{\sigma }}$ ${\displaystyle C_{\sigma }\cup C}$ ${\displaystyle C_{\sigma }}$ ${\displaystyle C}$ ${\displaystyle C}$

Is the source language, often with designated when the signature is to be specified, is the constant expansion around with or referred. ${\displaystyle L}$ ${\displaystyle L_{\sigma }}$ ${\displaystyle \sigma }$ ${\displaystyle C}$ ${\displaystyle L(C)}$ ${\displaystyle L_{\sigma }(C)}$

Applications

Completeness Theorem

Henkin's proof of Gödel's completeness theorem constructs a model for every consistent set of statements in a language of first-order predicate logic. An essential step is the addition of a new constant for each statement of the form . Each of these new constants whose entirety is called, acts as an example of an item that the existence statement fulfilled accurately obtained by adding the statements at the appropriate consistent set of statements again a consistent set of propositions from which one the existence of a model in the language show can. This model is then also in the language a model for the initially given consistent set of statements, because this does not contain the new constants from yes. This is Henkin's proof of the theorem of completeness. ${\displaystyle L}$ ${\displaystyle c_{\varphi }}$ ${\displaystyle \varphi =\exists x\psi (x)}$ ${\displaystyle c_{\varphi }}$ ${\displaystyle C}$ ${\displaystyle \varphi }$ ${\displaystyle \exists x\psi (x)\rightarrow \psi (c_{\varphi })}$ ${\displaystyle L(C)}$ ${\displaystyle L}$ ${\displaystyle C}$

Existence of great models

Let it be a set of statements of a language of first order predicate logic. If for every natural number there is a model of whose cardinality is greater than this natural number, then there are models of arbitrarily large cardinality. This statement, also referred to as the upward version of the Löwenheim-Skolem theorem, can be proven very easily by means of constant expansion as follows. For the given cardinal number choose constants . The given set of statements remains consistent if one adds the statements for every two , because every finite subset of the set of statements thus expanded only contains finitely many of the inequalities ; for this there are models according to prerequisites and the compactness theorem then provides a model for in the language . Every such model is also a model in the language and, because of the power of the added set of constants, has at least the power with which the statement is proven. ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle \kappa }$ ${\displaystyle c_{\alpha },\alpha <\kappa }$ ${\displaystyle \neg c_{\alpha }\equiv c_{\beta }}$ ${\displaystyle \alpha <\beta <\kappa }$ ${\displaystyle \neg c_{\alpha }\equiv c_{\beta }}$ ${\displaystyle T\cup \{\neg c_{\alpha }\equiv c_{\beta }|\alpha <\beta <\kappa \}}$ ${\displaystyle L(\{c_{\alpha }|\alpha <\kappa \})}$ ${\displaystyle L}$ ${\displaystyle \kappa }$

Individual constants

If a model is for a language with a set of carriers , it can be useful to have a constant for each individual . The constant expansion resulting from the addition of all constants is denoted by. The model in the current formulas of the language are then exactly the -statements in the model , if each individual constant through is interpreted. This perspective comes into play with the diagram method. ${\displaystyle {\mathcal {M}}}$ ${\displaystyle L}$ ${\displaystyle M}$ ${\displaystyle m\in M}$ ${\displaystyle c_{m}}$ ${\displaystyle c_{m}}$ ${\displaystyle L(M)}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle L}$ ${\displaystyle L(M)}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle c_{m}}$ ${\displaystyle m}$

Individual evidence

1. ↑ Wolfgang Rautenberg : Introduction to Mathematical Logic. A textbook . 3rd edition Vieweg + Teubner, Wiesbaden 2008, ISBN 978-3-8348-0578-2 , section 3.2.
2. ^ Heinz-Dieter Ebbinghaus , Jörg Flum, Wolfgang Thomas: Introduction to mathematical logic . 4th edition. Spectrum Akademischer Verlag, Heidelberg 1998, ISBN 3-8274-0130-5 , Chapter V: The Completeness Sentence .
3. ^ Chen Chung Chang, Howard Jerome Keisler : Model Theory (Studies in Logic and the Foundations of Mathematics, Volume 73). Elsevier Science Amsterdam 1990, ISBN 0-444-88054-2 , Corollary 2.1.6.