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# System prepended metadata

title: '#MTB'

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MTfB
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Typos / minor things
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ix line 13 of the 2nd paragraph: "texts on ~~in~~ model theory"

x line 3: 1980s (without an apostrophe)

x1 line 9 from bottom: 1960s (remove apostrophe)

p6 last line of section 1.4: "and its domain is its reduct" - maybe clearer to say "one of its reducts" or "its simplest reduct".

p11, first line of "2.3 Satisfaction" section: "**the** language" instead of "**a** language"

p12 6th from bottom: "induction on **the** complexity" (add the)

p20 3rd line: should be "and $b_1, ..., b_m$ in $M$ " instead of $b_1, ..., b_n$

p26 Thm 4.13 "that is **parametrically** definable" (add parametrically)

p26 Thm 4.23 proof: $f(M^n \setminus X) \subseteq M^n \setminus X$ (add exponent n)

p27 Section 4.3. line 7: replace **,** with **.** at end of line.

p30 Exercise 4.30: "every **positive** rational" (add positive)

((Page 33. Delete Proposition 5.7 and the sentence that precedes it.))

p31 1st line after Def 5.1. 1950s (no apostrophe) 

p35 Proof of Thm 5.12, end of 2nd line: **an** instead of **and** 

((Page 35. Proposition 11 should say: For each $n>0$, there are exactly $n!$ complete $n$-types of $n$-tuples with distinct elements that are realized in $(\Q,<)$.))

p40 Proof of Thm 6.4. line 13: should be $f_k(a_1)$ etc instead of $f(a_1)$ etc

p41 line 2: should be $\varphi_k$ instead of $\varphi_j$

p41 line 14 from bottom: replace "that" with "such that" or "for which" 

((Page 43. Three lines above Example 6.8: Proposition 6.10.))

p43, 3rd to last line: ...on **the** complexity...

p44, line 6: for some $b$ in $M$ (instead of $\mathfrak{M}$)

p44, at the beginning of section 6.4.: Theorem **6.4** (instead of 6.3)

p49, line 9 from bottom: ... the atomic formulas in by...: remove **in**

p49, line 8 from bottom: space missing after "i.e."

((Page 49. Line -2 at the bottom: psi should have been \psi.))

p51, line 8: but **that** it does not - remove "that" or even better replace by "clearly" 

p57, line 8: indic**a**tion (missing "a")

p58, 1st line of 8.3: The ~~are~~ oddities (remove "are")

p61, line 5 after Def 8.9: structure that (missing space)

p65, Section 8.7 line 3: primes ~~it~~ has

Comments / Notes to self
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p6 line 4 from bottom "...is the disjoint union..." - not everyone will know exactly what this is, is there even a standard definition? Often seen: $V \times \{0\} \cup F\times  \{1\}$.

p11 "the variable $x_1$ is *quantified* by $\exists$": (1) maybe "... is **said to be** *quantified* ..." as we're introducing it here (2) could be useful to also mention the alternative bound "... *quantified* (or *bound*) by $\exists$"

p14: The notations $\mathcal{R}^{\mathfrak{M}}$ and $c^{\mathfrak{M}}$ are used, but later on (page 23) we switch to  $\mathcal{R}_{\mathfrak{M}}$ and $c_{\mathfrak{M}}$

p17 3rd line: "formula $\varphi(x_1, .., x_n)$ **of** the language" - this "of" is used throughout the text, imo "in the language" sounds better

p17 7th line: it would be clearer to first define

$$
    \varphi(x) = (x = y_1) \vee ... \vee (x = y_m) 
$$

p17 after DEFINITION 3.9 - re o-minimal: would be useful to note that the complement of such sets is also o-minimal (which I believe it is?).

p23, section 4.1. line 4: instead of "one-to-one and onto function", why not just write "bijection"? ... ok now I see this is consistent throughout the text

p24 Proof of thm 4.6.: constants have received a special treatment before, but here the proof only considers relations. Some may ask themselves (as I did) why did we single out these constants, when we don't really need them. Might be useful to mention that having constants is just a useful notation, but as they are just special cases of relations, they need not be considered in proofs.

p34. line 4 "to types **as defined above** as follows" - the "as defined above" is not really necessary as it was defined just 2 lines above :-)

p34 Exercies 5.9. maybe two small gaps: (1) "complete theory" not defined before as far as I can tell (2) "isolated type in one model of T" also not defined before. Both are obvious though.

p34 beginning of section 5.3: it is not immediately clear why one should consider only types like $tp(a,b,c)$ where $a$, $b$ and $c$ are pairwise distinct. What about $tp(a,b,a)$ or $tp(a,a,b)$ etc? Maybe note that these last two types are essentially equivalent to
$tp(a,b)$ - cf "Notes on types in ordered structures" below

p35/36 Theorem 5.12 (isomorphism of countable densely linearly ordered sets without endpoints) - note that the back-and-forth method was not Cantor's original proof. Cf [wikipedia](https://en.wikipedia.org/wiki/Back-and-forth_method)

p41: **!??** "the set of sentences $\varphi(a)$ such that $\mathfrak{M} \models \varphi(a)$, where a is a constant symbol added to the language". It seems that $\varphi(a)$ is not a "legal" formula, so what does it stand for? Is it $\varphi(x) \land (x = a)$ ? Not possible as this has a free variable. So what is it then? Is it $\exists x(\varphi(x) \land (x = a))$ -- ok this was answered by RK

p49, Th 7.4 "We will only prove the compactness theorem for countable languages" - it might be useful to at least outline how the proof would change for arbitraryily large cardinalities, as we use that later (p.53, end of proof of Corollary 7.10)

- we require a well-ordering of the language
- the need as many constant synbols $c_i$ as there are relations in the language 
- we can then well-order the sentences $\psi_i$ (explain why we can do this, and why the cardinality of these sentences is the same as the cardinality of the language ??)
- we can then use transfinite induction, first for defining $\varphi_i$, then for defining $T_i$

The rest of the proof should be almost the same (to be confirmed)

p52 7.2.1 line 5. "It follows directly from defintions ..." is somewhat confusing. How about: "**We will show that** it follows directly from definitions that if $\mathfrak{M} \models T$ then, **in some precise sense**, $\mathfrak{M} \prec \mathfrak{N}$. ~~Here we~~ **We will** take advantage ..."


Writing formulas
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Integrating the successive simplifications of notation was a challenge - not intuitively but on the level of formal definitions. Treatment of constants was also quite confusing initially.


Notes on types in ordered structures (cf 5.3.)
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We wish to classify the types of triples in a linearly ordered structure.

(1) **This is what's in the book** Let's start with $(a,b,c)$ in which all three elements are distinct. How does $tp(a,b,c)$ relate to, say, $tp(b,a,c)$?

For a formula $\varphi(x_1, x_2, x_3)$, define $\varphi^*(x_1, x_2, x_3) = \varphi(x_2, x_1, x_3)$. Then $\varphi \in tp(a,b,c)$ iff $\varphi^* \in tp(b,a,c)$. (BUT are we sure there will be no clashes with bound variables? to be verified)

This means that given $tp(a,b,c)$, we can directy obtain the type of any permutation of $(a,b,c)$ by simply shuffling free variables around. It is therefore enough to consider triples such that $a < b < c$.
    
(2) But what if we have a triple in which two of the elements are the same, for example $(a,b,a)$, with $a$ and $b$ distinct?

First, note that $\psi \in tp(a, b)$ implies $\psi \in tp(a,b,a)$, because $\psi$ does not have $x_3$ as a free variable.

For a formula $\varphi(x_1, x_2, x_3)$, we constuct $\varphi^*(x_1, x_2)$ by taking $\varphi$ and replacing all occurrences of the free variable $x_3$ by $x_1$ (this may require renaming bound variables if/where necessary). Then $\varphi \in tp(a,b,a)$ iff $\varphi^* \in tp(a,b)$.

What this means is that there is a one-to-one relationship between $tp(a,b,a)$ and $tp(a,b)$.

(3) In the case of $(a,a,b)$, with $a$ and $b$ distinct, we cannot directly apply the method above. We first need to "push distinct values to the front" as follows.

For a formula $\varphi(x_1, x_2, x_3)$, define $\varphi^*(x_1, x_2, x_3) = \varphi(x_1, x_3, x_2)$. Then $\varphi \in tp(a,a,b)$ iff $\varphi^* \in tp(a,b,a)$.

Next we apply the procedure in (2) to find that $tp(a,a,b)$ is also essentially similar to $tp(a,b)$.

(4) Finally, consider the triple $(a,a,a)$.

Clearly, $\psi \in tp(a)$ implies $\psi \in tp(a,a,a)$ as $\psi$ does not have free variables $x_2$ and $x_3$.

For a formula $\varphi(x_1, x_2, x_3)$, we constuct $\varphi^*(x_1)$ by taking $\varphi$ and replacing all occurrences of the free variables $x_2$ and $x_3$ by $x1$ (again renaming bound variables where necessary). Then $\varphi \in tp(a,a,a)$ iff $\varphi^* \in tp(a)$.

Note on variable naming (to simplify free variable replacement)
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Take an arbitary formula (with the usual variables $x_1$, $x_2$, ...) and systematically replace all quantified/bound variables by $y_1$, $y_2$, ...

This has two effects: (1) free variables are clearly visible (2) there is no repetition of bound variables and (3) replacement of free variables becomes straightforward. Take for example:

$P(x_1, x_2) \land \exists x_2 Q(x_1, x_2) \land \exists x_2 R(x_1, x_2) \land S(x_3)$

Here $x_2$ is free only in the first part of the formula. The same variable $x_2$ is also bound to two different quantifiers, so we have three different "instances" of $x_2$. Even though this formula is well-formed, this can be confusing. Also, if we wanted for example switch $x_2$ and $x_3$, we would have to keep track of where these variables occur in free and quantfied form. Rewriting it yields:

$P(x_1, x_2) \land \exists y_1 Q(x_1, y_1) \land \exists y_2 R(x_1, y_2) \land S(x_3)$

Now we can easily switch $x_2$ and $x_3$.

(The variables $y_i$ can formally be seen as variables $x_{k(i)}$ where the indices $k(i)$ do not overlap with the indices of the free variables)