Loading posts...
  • Exercises #4 – Equalitarian Theories

    These were pulled from Exercises for [latex]\S5[/latex] of Chapter I in Bourbaki’s set theory book. (Assume [latex]\mathfrak{T}[/latex] has no constants, i.e. no letters in explicit axioms; thus we can abuse I.2.3.C3 without repercussions) Exercise 1.Solution.Exercise 1. Show the relation [latex]x=y[/latex] is functional in [latex]x[/latex] in [latex]\mathfrak{T}[/latex]. Solution. Denote the relation [latex]x=y[/latex] by [latex]R[/latex]. Let [latex]a[/latex] and…
  • Exercises #3 – Quantified Theories

    These were pulled from Exercises for [latex]\S[/latex]3 and Exercises for [latex]\S[/latex]4 of Chapter I in Bourbaki's set theory book. Exercise 1.Solution.Exercise 1. Let [latex]A[/latex] be a relation in logical theory [latex]\mathfrak{T}[/latex]. If [latex]A\iff (\mbox{not } A)[/latex] is a theorem in [latex]\mathfrak{T}[/latex], show that [latex]\mathfrak{T}[/latex] is contradictory. Solution. The negation of [latex]A\iff (\mbox{not } A)[/latex] is…