First order language

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Needed for set theory

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Tasks:

1. Expand on the logical connective symbol caution.

There are notes on this topic

Definition

A first order language, [ilmath]\mathcal{L} [/ilmath] consists of two types of symbols, non-logical and logical, these are described in the tree below:

• Non-logical symbols - these are the same for all first order languages
  • [ilmath]V[/ilmath] - The set of (at most countably many, possibly empty) variable symbols: [ilmath]x_1,x_2,\ldots,x_n,\ldots[/ilmath]
  • [ilmath]C[/ilmath] - The set of logical connective symbols. Caution:Not all of these are needed, you can write some in terms of others
    1. [ilmath]\neg[/ilmath] - logical not
    2. [ilmath]\wedge[/ilmath] - logical and
    3. [ilmath]\vee[/ilmath] - logical or
    4. [ilmath]\rightarrow[/ilmath] - logical implication, "if ... then ..."
    5. [ilmath]\leftrightarrow[/ilmath] - logical equivalence (AKA: if and only if)
  • [ilmath]Q[/ilmath] - The set of quantifier symbols. Caution:Given [ilmath]\neg[/ilmath] you can define [ilmath]\forall x(A)[/ilmath] as [ilmath]\neg(\exists x(\neg(A)))[/ilmath] or define [ilmath]\exists x(A)[/ilmath] as [ilmath]\neg(\forall x(\neg(A)))[/ilmath]
  • [ilmath]E[/ilmath] - The set containing the equality symbol. We will use [ilmath]\doteq[/ilmath] for this (to separate it from equality in the meta-language)
  • [ilmath]B[/ilmath] - The set of brackets, that is "(" and ")".
    • AKA: [ilmath]P[/ilmath] - for "parentheses", but you know it's BODMAS not "PODMAS".
• Non-logical symbols - these vary from language to language
  • [ilmath]\mathscr{L}_c[/ilmath] - the set of (possibly zero, at most countably many) constant symbols, [ilmath]c_1,c_2,\ldots,c_n,\ldots[/ilmath].
  • [ilmath]\mathscr{L}_f[/ilmath] - the set of (possibly zero, at most countably many) function symbols.
    • Each function has an arity, and we write [ilmath]ft_1t_2\cdots t_m[/ilmath] for an [ilmath]m[/ilmath]-ary function (here the [ilmath]t_i[/ilmath] are terms)
      • Here [ilmath]m\ge 1[/ilmath] Caution:This is disputed, Kunen's Set Theory gives an example of [ilmath]0[/ilmath]-ary predicates and functions!
  • [ilmath]\mathscr{L}_P[/ilmath] - the set of (possibly zero, at most countably many) predicate symbols, [ilmath]P_1,P_2,\ldots,P_n,\ldots[/ilmath].
    • Each predicate has an arity and we write [ilmath]Pt_1t_2\cdots t_m[/ilmath] for an [ilmath]m[/ilmath]-ary predicate symbol. Here the [ilmath]t_i[/ilmath] are terms
      • Here [ilmath]m\ge 1[/ilmath] Caution:This is disputed, see caution above

See next

• Term - the set of all terms is denoted [ilmath]\mathscr{L}_T[/ilmath]
• Formula - the set of all formulas is denoted [ilmath]\mathscr{L}_F[/ilmath]
  • Sentence - a special kind of formula

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TODO: Add more

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