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sponsoredwalk
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Every so often I return to the idea of structures as used in logic & clean up my ideas
about them, here is my latest attempt, four questions based on the short summary
below, would really appreciate any help from you guys.
A structure is a triple of the form = (F, σ, I).
F is a set of objects.
σ is the signature, a set of function &/or relation symbols with no meaning ascribed to them.
I is an interpretation "function" that takes elements of the signature & gives them meaning.
If σ = {+,•,0,1} then:
I(+) : F × F → F | (a,b) ↦ +(a,b) = (a + b);
I(•) : F × F → F | (a,b) ↦ •(a,b) = (a • b);
I(0) ∊ F;
I(1) ∊ F.
That is my summary of the wiki page on structures.
I'm trying to reconcile this material with the following:
1: Ebbinghaus in Mathematical Logic describes a structure as something of the form
= (F,+,•,0,1)
I can see that = (F, σ, I) is like a shorthand for Ebbinghaus' explicit way of
showing what's in the structure but there is no interpretation function in Ebbinghaus,
what's going on there?
2: If we're working with = (F, σ, I) what happens to all of the definitions of
relations & structures that are commonly found in most textbooks?
For example, A (binary) relation is defined also as a triple of the form (A,B,G)
where G ⊆ A × B. This triple is nothing like (F, σ, I)! The question here is not
about the apparent contradiction between = (F, σ, I) & (A,B,G) [check next question],
I am asking about the way something like (A,B,G) fits into = (F, σ, I). So on the
one hand σ = {+,•,0,1} is described as a set of function symbols lacking meaning but
on the other hand things like + & • are described as sets. (A,A,+) describes
(A,A, + ⊆ A × A), so + is a graph.
3: What about the apparent contradiction in notation, here is a triple rigorously
founded in logic, = (F, σ, I), and here is a triple given with no justification
(A,B,G), I'm assuming that (A,B,G) is really some kind of (F, σ, I) in disguise?
4: Why are 0 & 1 in the signature? Surely 0 & 1 are in all sets you commonly work with
regardless? My question might be clearer by asking about symbols such as ℕ ℤ ℙ ℚ ℝ etc...
Is ℕ = (N, σ, I) or N = (ℕ, σ, I)? Think about that, if ℕ = (N, σ, I) then I can see that
the set N of objects does not contain 0 or 1 & you need some function to put them in there
but if N = (ℕ, σ, I) then 0 & 1 are already in the set of objects ℕ & the structure N has
some extra 0 & 1. I don't really understand what's going on here.
(5: Do you have, or can you find, a boldface F, i.e. , that doesn't create massive
spaces between lines? Preferably an alphabet of html-friendly boldface letters!).
about them, here is my latest attempt, four questions based on the short summary
below, would really appreciate any help from you guys.
A structure is a triple of the form = (F, σ, I).
F is a set of objects.
σ is the signature, a set of function &/or relation symbols with no meaning ascribed to them.
I is an interpretation "function" that takes elements of the signature & gives them meaning.
If σ = {+,•,0,1} then:
I(+) : F × F → F | (a,b) ↦ +(a,b) = (a + b);
I(•) : F × F → F | (a,b) ↦ •(a,b) = (a • b);
I(0) ∊ F;
I(1) ∊ F.
That is my summary of the wiki page on structures.
I'm trying to reconcile this material with the following:
1: Ebbinghaus in Mathematical Logic describes a structure as something of the form
= (F,+,•,0,1)
I can see that = (F, σ, I) is like a shorthand for Ebbinghaus' explicit way of
showing what's in the structure but there is no interpretation function in Ebbinghaus,
what's going on there?
2: If we're working with = (F, σ, I) what happens to all of the definitions of
relations & structures that are commonly found in most textbooks?
For example, A (binary) relation is defined also as a triple of the form (A,B,G)
where G ⊆ A × B. This triple is nothing like (F, σ, I)! The question here is not
about the apparent contradiction between = (F, σ, I) & (A,B,G) [check next question],
I am asking about the way something like (A,B,G) fits into = (F, σ, I). So on the
one hand σ = {+,•,0,1} is described as a set of function symbols lacking meaning but
on the other hand things like + & • are described as sets. (A,A,+) describes
(A,A, + ⊆ A × A), so + is a graph.
3: What about the apparent contradiction in notation, here is a triple rigorously
founded in logic, = (F, σ, I), and here is a triple given with no justification
(A,B,G), I'm assuming that (A,B,G) is really some kind of (F, σ, I) in disguise?
4: Why are 0 & 1 in the signature? Surely 0 & 1 are in all sets you commonly work with
regardless? My question might be clearer by asking about symbols such as ℕ ℤ ℙ ℚ ℝ etc...
Is ℕ = (N, σ, I) or N = (ℕ, σ, I)? Think about that, if ℕ = (N, σ, I) then I can see that
the set N of objects does not contain 0 or 1 & you need some function to put them in there
but if N = (ℕ, σ, I) then 0 & 1 are already in the set of objects ℕ & the structure N has
some extra 0 & 1. I don't really understand what's going on here.
(5: Do you have, or can you find, a boldface F, i.e. , that doesn't create massive
spaces between lines? Preferably an alphabet of html-friendly boldface letters!).