I am currently struggling with a homework question to do with first order logic structures. The problem is as follows:
This question concerns the structure $(ℝ,+,*,-,-,<, \leq ,| |, f, 0 ,1) $ where $+,*,-,-$ are the usual arithmetic operators (addition and multiplication are binary, and there is a unary minus as well as binary minus), $<,\leq$ are the usual binary order relations, $||$ is a unary function for absolute value, $f$ is a unary function for some other $f:ℝ$ → $ℝ$ and $0$ and $1$ are constants.
express as a statement in this language: $f$ is continuous at $0$
I have expressed this as $ ∃x(f(0)=x) ∧ ∀z∀y(0< y → ∃w(0<w ∧ 0<|z|<w →|f(z)-f(0)| < y )) $
what this means is that an $x$ exists such that $f(0)$ is $x$ and the limit of $f(x)$ as $x$ tends to $0$ exists, I did this by adapting the statement: the limit of f(x) as x tends to a is L if:
for every number $e>0$, the exists $d> 0$ such that $|f(x)-L|<e$ whenever $0<|x-a|<d$
I think this part is right, however then ext part of the question is what confuses me, it says:
the statement should be correct in the given structure, give also an example of some structure that is based on ℝ or ℚ as an ordered field in which the statement is false.
Would it be sufficient to just use the same structure but exclude the $<$ leaving only the $\leq$ symbol so that the above statement of f being continuous at 0 can't be written?
Thanks
You're confused about what "structure" means here. I'll talk through a similar problem and hopefully that'll help you understand.
Say you have a logical sentence or set of consistent sentences, $T$. For example, maybe $T$ asserts the properties of a dense linear order without endpoints. A model of $T$ is a set for which $T$ is true. So the real numbers, the rational numbers, and the irrational numbers are all examples of a model of $T$.
When we talk about a structure we are talking about a set, like the above examples, in which a set of sentences can be true or false. We can't call it a model because models are what you get when they turn out to be true. Most mathematical objects you're used to seeing are structures.
In our example, structure that are not models, i.e. ones where the sentences $T$ don't hold, include the rational numbers with denominator $<10$, the unit interval, and the complex numbers.