EquatReps092310 - EQUATIONAL REPRESENTATIONS by Harvey M...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
EQUATIONAL REPRESENTATIONS by Harvey M. Friedman* Ohio State University [email protected] http://www.math.ohio-state.edu/%7Efriedman/ September 23, 2010 NOTE: THIS IS AN ADVANCED DRAFT THAT NEEDS SOME POLISHING BEFORE SUBMISSION FOR PUBLICATION. 1. Preliminaries: N, N, ℘℘ N. 2. Fundamental Lemma Concerning Clones. 3. From ( f1,...,fk)( x1,...,xp)( ϕ ) To ( binary f)( x,y)( ϕ ). 4. From N, N, ℘℘ N to N N, ( N N). 5. From N N, ( N N) To Binary Sentences Over ( N N,...). 6. From Binary Sentences Over ( N N,...) To Binary Sentences Over ( N, ). 7. From Binary Sentences Over ( N, ) To Binary Sentences Over ( ,<). 8. From Binary Sentences Over ( ,<) To ( f: )( x,y )(s = t). 1. Preliminaries: N, N, ℘℘ N. We begin by presenting the language L(N, N, ℘℘ N). This is the standard language for presenting third order sentences, using its intended interpretation. There are variables n i , i 1, over N. Variables x i , i 1, over N. Variables A i , i 1, over ℘℘ N. The terms are defined inductively by Each n i is a term. 0 is a term. If s,t are terms, then S(t), s+t, s t are terms. The atomic formulas are x i = x j . A i = A j . x i A j . t x i , where t is a term. s = t, where s,t are terms.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Formulas of L(N, N, ℘℘ N) are obtained using connectives and quantifiers in the usual way. Their intended meaning is obvious. The Σ 1 formulas of L(N, N, ℘℘ N) are the formulas without any quantifiers over ℘℘ N. The Σ 2 1 formulas of L(N, N, ℘℘ N) are the formulas of L(N, N, ℘℘ N) are the formulas that begin with zero or more existential quantifiers over ℘℘ N, followed by a Σ 1 formula of L(N, N, ℘℘ N). We use ZC for Zermelo set theory with the axiom of choice, which we will take to be ZFC with Replacement omitted. The Σ 21 sentences of L(N, N, ℘℘ N) are well known to be very expressive. For example, the continuum hypothesis is provably equivalent to a Σ 2 1 sentence, over ZC. Throughout the paper, we take the Σ 2 1 sentences to be the Σ 2 1 sentences of L(N, N, ℘℘ N). Here is the main result of this paper. THEOREM 7.5. The continuum hypothesis (more generally, every Σ 21 sentence) is provably equivalent, over ZC, to a sentence of the form ( f: )( x,y )(s = t) where s,t are terms in f,x,y,+, ,0,1. Here are some other results of significance. THEOREM 6.1. The continuum hypothesis (more generally, every Σ 2 1 sentence) is provably equivalent, over ZC, to a sentence of the form ( f: 2 )( x,y )( ϕ ) where ϕ is quantifier free in f,x,y,<. THEOREM 5.1. The continuum hypothesis (more generally, every Σ 21 sentence) is provably equivalent, over ZC, to a sentence of the form ( f:( N) 2 N)( x,y N)( ϕ )
Image of page 2
where ϕ is quantifier free in f,x,y, .
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern