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user400188
user400188
9:24 AM
@user21820 I'll wait until I am taught the different definitions before I return to the question then.
 
 
5 hours later…
Threnody
Threnody
1:54 PM
Do we need SOL to talk about Dedekind cuts? As I understand it, some more reals (apart from N and Q) are obtained from Dedekind cuts which involve partitioning Q into two subsets for each new real, so we need quantification over all the subsets of Q to get all the reals given this way. Do the Dedekind cuts really give us 'all' the other reals? What about uncomputable reals?
 
 
2 hours later…
user21820
user21820
4:04 PM
@Threnody I'm not sure how much of your inquiry can be answered at this point. Whatever it is, you must remember that all known choices of MS can be expressed in FOL. The very notion of dedekind cuts is something defined within MS, because after all if you have just a structure whose elements are real numbers, there are obviously no sets!
So yes to be able to talk about sets of reals you cannot be working within the structure with just reals.
But there are many questions you could potentially ask about even just the structure of the reals in the language of ordered fields, namely (R,0,1,+,·,<).
For one, what kinds of structures are models of the FOL theory of the reals?
 
user21820
user21820
4:18 PM
4
A: formalizing the theory of real numbers

user21820Since you're curious, here's a curious fact. The computable reals have exactly the same first-order theory as the 'real' reals. And for any real-world (engineering, physics, ...) application one needs (and can manipulate) only computable reals. So arguably we don't need anything more than the fir...

29
A: Is the real number structure unique?

user21820To paint a more complete picture, you are right in that an axiomatization may very well have no model. An axiomatization is meaningless if nothing satisfies it. But if we can prove that there is a model, and the axioms are the only properties we care about, then we can happily work within the axi...

Those two posts were written for a higher-level intended audience, so it's okay if you don't understand half. Let's just say that every countable FOL theory has a countable model, so you automatically know that the theory of the reals has a countable model. Th(R) is also known as the theory of real closed fields, and it includes theorems such as "∀x ( x>0 ⇒ ∃y ( y·y = x ) )".
You can pin down R relative to MS, because there is only one unique ordered field satisfying the (second-order) completeness axiom, up to isomorphism. But this does not contradict the above fact because a countable model of MS (if it exists) will of course see only a countable set of reals.
 
user21820
user21820
4:54 PM
 
Threnody
Threnody
5:11 PM
@user21820 I'll give them a shot, thank you :)
 
user21820
user21820
You're welcome!
 
Threnody
Threnody
"the first-order induction schema and the single second-order induction axiom" is it always the case that we can grab a schema and convert it to an equivalent axiom by 'one-upping' the logic's order? Vague question I think... but I do (maybe incorrectly) remember you showing me how Fitch style can 'convert' the comprehension schema into a 'Given x in S' axiom... I might have forgotten the details however :/ it had something to do with Hilbert's deductive system I believe
Also... I noticed, whenever you involve fields like F or R, you always explicitly state the identities
Like, (R, 0, 1, ...)
Is this a style preference or is there a reason for it?
Thank you for your patience, I must go for a while :)
 
user21820
user21820
5:50 PM
@Threnody That phrase was about the difference between the first-order induction schema and the second-order induction axiom when under full second-order semantics. I don't really want to talk about that until you actually get a proper understanding of FOL first.
Roughly speaking, full second-order semantics means that a structure is defined by a domain and interpretations of the symbols, and the second-order quantifiers are interpreted (within MS) as quantifying over subsets of the domain. So truth in a second-order structure under full semantics is determined by the objects in MS itself, and not in terms of the structure alone.
And unlike FOL, there is no way to systematically enumerate sentences that are true in every full semantics model of a given SOL (second-order logic) theory.
Because for example PA with the second-order induction axiom is satisfied by a unique full semantics model up to isomorphism, so all sentences that are true in that model include all the true arithmetical sentences (as viewed within MS), which the incompleteness theorems already tell us cannot be enumerated.
 

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