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The meaning of definition written in the form ... is...

Mathematics Asked by Jsi23484 on January 1, 2022

I am studying mathematics logic and i am now criticising all the things…

I met some definitions which are in the form of … is ….

For example, A construction sequence for an expression $alpha$ is a finite sequence $alpha_1,…alpha_n=alpha$ such that each $alpha_i$ is member of core set or it is a result of applying some $fin P$ to $alpha_i,…alpha_j$.

My question is, is it true that the set of construction sequence for $alpha$ is equal to the set of finite sequence $alpha_1,…alpha_n=alpha$ such that each $alpha_i$ is member of core set or it is a result of applying some $fin P$ to $alpha_i,…alpha_j$?

Moreover, are there any formal way to construct definition in mathematics?

One Answer

is it true that the set of construction sequence for 𝛼 is equal to the set of ...

Yes, trivially -- they are equal by definition, because the same set is being talked about. There is the same written to the left and right, it just looks different.

A definition is just syntactic sugar for the meta language: It means that instead of writing "a finite sequence $alpha_1,...alpha_n=alpha$ such that each $alpha_i$ is member of core set or it is a result of applying some $fin P$ to $alpha_i,...alpha_j$", we can write "a construction sequence for $alpha$". There is nothing deeply set-theoretic going on behind it: A definition is just inventing a new word for some mathematical concept to use as a convenient abbreviation to refer to said concept in texts. The structure underlying the concept already existed before inventing a special word for it, and writing up a definition doesn't create a new kind of entity nor does it perform any sort of set-theoretic operation on objects: It's just giving that thing a short name (the definiendum) so we don't have to spell out its exact description (the definiens) every time we want to mention it in a text; the short name and the long description refer to the same one thing.

If you want to define it formally, you can say something like a definition being an expression in the meta language, "A := B", whose semantics is that the extension of A is identical to the extension of B (and the extension of B again is defined by what the extension of the word "sequence" etc. is, until we come down to words for concepts that can no longer be decomposed into already defined concepts any further, such as the concept of a set). This doesn't entirely capture the idea of a definition; we rather need some kind of operational semantics with meaning assignment rather than merely a statement about equality, but you get the idea.

Answered by lemontree on January 1, 2022

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