## Non-standard Models of Arithmetic 19

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.

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## 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.

Filed under Analysis, Groups, Topology

## 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:

1. A is a T-standard model of PA.
2. A is a recursively saturated model of PA+ΦT.

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## 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.

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## 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

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## Non-standard Models of Arithmetic 15

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.

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## Algebraic Geometry Jottings 18

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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:

$\text{res}_x(E,F) = a_m^n(y) b_n^m(y) \prod_{i=1}^m\prod_{j=1}^n (u_i-v_j)$  (1)

implies Fact 3: