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In linear algebra we learn about the idea of a set of 'polynomials' would this set be equivalent to a normal set such as the set of real numbers? The idea of sets (in my understanding) is that they can contain Mathematical objects (numbers, functions, other sets etc). Do we consider expressions as Mathematical objects? In which case is it more correct to say the following in terms of equality:

'The mathematical object that $x+1$ represents is the same as the mathematical object that $x+2-1$ represents'?

Instead of:

'$x+1$ and $x+2-1$ are the same mathematical object'

If expressions are objects in their own right? I would generally consider them 'syntactic objects' but the fact we can form sets with them suggests they are Mathematical Objects.

Edit: I have learnt that in dealing with Polynomials we are dealing with mappings based on indeterminates, if we can have an expression as part of a set, something like '$x+1$ is an element of A' is ambiguous, as are we referring to the expression or it's value? Is it possible to put an expression in a set?

user37577
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    Yes, we can indeed think of expressions as objects in their own right, with the "represents" relation being very different from "is the same as." Basically all of mathematical logic involves this conceptual shift, but in particular model theory is more-or-less entirely about this. I'll add a detailed answer when I have time, but I just wanted to quickly say that this is indeed a very real thing. (Meanwhile you may be interested in the latter part of this old answer of mine.) – Noah Schweber Aug 06 '22 at 17:22
  • So it's not problematic to have a set of expressions? How does this work, in terms of how to we define the set using set builder notation or something. – user37577 Aug 06 '22 at 17:37
  • You might also find Gödel numbering interesting. It is a way of mapping between formulae and natural numbers. – kc9jud Aug 06 '22 at 17:39
  • By the way, in your examples, the mathematical objects in question are functions. So you could say "the function that $x+1$ represents is the same as the function that $x+2-1$ represents". – Lee Mosher Aug 06 '22 at 18:50
  • I am not well versed in set theory, but I do consider expressions such as $p(x)$ for some polynomial $p(x)$ to be mathematical objects. I often understand them as level sets, e.g. ${x : p(x) = 0}$. Then there become many interesting questions about the set ${x : p(x) = 0}$ "as a mathematical object." – Alex Ortiz Aug 06 '22 at 19:22
  • @AlexOrtiz my issue with it is that does {1,2,3} mean that the set contains the NUMBERS 1,2 and 3 or the expressions '1','2' and '3'? – user37577 Aug 06 '22 at 19:28
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    In response to your response to Alex: what are the “numbers” $1,2,3$? Do you here consider them as elements of $\Bbb N$, typically “constructed” by Von Neumann’s empty-set construction, or as elements of $\Bbb Z$, typically constructed as equivalence classes of ordered pairs of naturals, or as elements of $\Bbb Q$, equivalence classes of certain pairs in $\Bbb Z$, or as elements of $\Bbb R$, (these are all my favourite constructions, not the only ones) equivalence classes of Cauchy sequences of rationals? Et cetera. I’m not versed in foundations either, but: does the question really matter? – FShrike Aug 06 '22 at 19:41
  • @FShrike what I'm saying is that, if 'expressions' can be in sets, than we need to make it clear if its the 'expression' or it's value that's in the set. – user37577 Aug 06 '22 at 20:05
  • What is an “expression”? This is not a term that has a precise definition (unless perhaps if someone defines it precisely in some particular context). When I’m trying to be perfectly rigorous, I avoid using the term “expression” entirely. – littleO Aug 08 '22 at 09:13

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Yes, expressions are considered mathematical objects. Examples of math objects include real numbers, compmex numbers, sets, functions, operators, algebraic structures, etc. Loosely speaking, mathematical objects are basically anything you can do math with. More formally speaking, they are objects that have a definition, follow certain rules and properties, and are the target for mathematical operations.

Accelerator
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  • When we define sets and rings of Polynomials would we consider them sets of expressions? – user37577 Aug 06 '22 at 18:36
  • If we're just talking about expressions and not functions, then yes because sets and rings of polynomials have elements. An expression is an arrangement of math symbols used to represent a mathematical object. For example, $5$, $6-1$, and $5x+6$ are expressions. An expression is basically the math version of a noun from English. – Accelerator Aug 06 '22 at 21:14
  • Is it not a little bit problematic to distinguish between the expression, and the number it represents? such as $x^2+2x+1 ∈ A$ could mean the value of '$x^2+2x+1$ is an element or the expression itself is an element of A, is it just contextual? – user37577 Aug 07 '22 at 11:51
  • I really don't think it's that deep. I'm pretty sure what you're asking is contextual too. @user37577 – Accelerator Aug 07 '22 at 20:00
  • Yes, I believe its not as deep as this, also in the case of 'polynomials' I was told that we can see them as sort of their own objects where two 'expressions' define the same polynomial if the coefficients are the same, its uncommon to find sets with 'expressions' outside this anyway. – user37577 Aug 07 '22 at 20:33
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    @user37577 "Is it not a little bit problematic to distinguish between the expression, and the number it represents?" No, it is not problematic, because in 99.9% of cases, it is clear from context what you mean. In the 0.1% of cases, you simply make sure to be explicit so as to avoid confusion. No problem anywhere. – 5xum Aug 08 '22 at 09:31