JB: Before we get into any proofs, I’d just like to marvel at Enayat’s Prop. 6, and see if I understand it correctly. I tried to state it in my own words on my own blog:
Every ZF-standard model of PA that is not V-standard is recursively saturated.
Author Archives: Michael Weiss
Non-standard Models of Arithmetic 19
Filed under Conversations, Peano Arithmetic
Bundles and Laplacians
I originally started this blog to make available various notes I’ve written over the years. (Justification: the internet hasn’t yet run out of space.) Herewith a very short note on principal and fiber bundles (small and medium formats), and a longer one on the Laplacian on the cube. Also Three takes on the tangent bundle.
Non-standard Models of Arithmetic 18
MW: To my mind, the heart of Enayat’s paper is Proposition 6 and Theorem 7, which combine to give Corollary 8.
Proposition 6: Every ZF-standard model of PA that is nonstandard is recursively saturated.
Theorem 7: Every countable recursively saturated model of PA+ΦT is a T-standard model of PA.
Corollary 8: The following statements are equivalent for a countable nonstandard model A of arithmetic:
- A is a T-standard model of PA.
- A is a recursively saturated model of PA+ΦT.
Filed under Conversations, Peano Arithmetic
Non-standard Models of Arithmetic 17
MW: I’d like to chew a bit more on this matter of Trued versus True. This Janus-feature of the Tarski legacy fascinated me from the start, though I didn’t find it paradoxical. But now I’m getting an inkling of how it seems to you.
Filed under Conversations, Peano Arithmetic
Non-standard Models of Arithmetic 16
MW: Ok, let’s plunge into the construction of Trued(x). The bedrock level: True0(x), truth for (closed) atomic formulas. Continue reading
Filed under Conversations, Peano Arithmetic
Non-standard Models of Arithmetic 15
The Truth about Truth
MW: A little while back, I noted something delicious about the history of mathematical logic:
- Gödel’s two most famous results are the completeness theorem and the incompleteness theorem.
- Tarski’s two most famous results are the undefinability of truth and the definition of truth.
Filed under Conversations, Peano Arithmetic
Algebraic Geometry Jottings 18
Up till now, I’ve been using power series to parametrize branches:
x(t) = a0+a1t+a2t2+ ⋯, y(t) = b0+b1t+b2t2+ ⋯
If the branch passes through the origin, then a0=b0=0. In the last post, we established Facts 4 and 5, assuming that y(t)=t for all branches, so x(t) = a0+a1y+a2y2+ ⋯.
Filed under Algebraic Geometry
Algebraic Geometry Jottings 17
At last we come to Kendig’s proof of Bézout’s Theorem. Although not long, it will take me a few posts to appreciate it in full.
Filed under Algebraic Geometry
Algebraic Geometry Jottings 16
The Resultant, Episode 5: Inside the Episode
The double-product form for the resultant:
implies Fact 3:
Filed under Algebraic Geometry
Algebraic Geometry Jottings 15
The Resultant, Episode 5 (The Finale)
Recap: The setting is an integral domain R, with fraction field K, and extension field L of K in which E(x) and F(x) split completely. E(x) and F(x) have coefficients in R. E(x) has degree m, F(x) degree n; we assume m,n>0. The main special case for us: R=k[y], K=k(y), so R[x]=k[x,y], and E and F are polynomials in x and y. As always, we assume k is algebraically closed.
Filed under Algebraic Geometry
Diagonal Argument 