I want to build an uncountable well-ordered set within the theory $Z^\textbf-$. So, I take $A=\omega$ (exists by infinity axiom) and define $$W:=\{(X,R)\in\mathcal{P}(A)\times\mathcal{P}(A\times A):\langle X,R\rangle\ \text{is a well-ordered set}\}$$

With this I consider the set $T:=W/\cong$ (where $\cong$ is isomorphism relation). Note that $W$ and $T$ are sets by power set axiom. So I define for each equivalence class $[x],[y]\in T$ the order

$$ [x]\leq_T[y]\Leftrightarrow \text{type}(x, R_x)\leq \text{type}(y,R_y) $$

Here, $R_x$ means the order of $x$ that make it belongs to $[x]$.

So, $T$ is well ordered by $\leq_T$ and it is uncountable.

Question: I don't pretty sure if I can build $\leq_ T$ without replacement axiom. Another doubt is, Can I take the $R_x$ without AC?

  • 1
    $\begingroup$ What is Z$^-$? (ZF without replacement is just "Z.") Also, "$\le_T$" is the standard symbol for Turing reducibility, so it might be better to use a different symbol here. $\endgroup$ – Noah Schweber Dec 3 '18 at 23:06
  • $\begingroup$ @NoahSchweber $Z^-$ means, $ZF$ without replacement and regularity. $\endgroup$ – Gödel Dec 4 '18 at 1:09
  • $\begingroup$ My edit was for a typo in the 1st line ("whitin"). And I wanted to find out what the code for $\cong$ is $\endgroup$ – DanielWainfleet Dec 4 '18 at 2:45
  • $\begingroup$ Another name for Regularity is Foundation $\endgroup$ – DanielWainfleet Dec 4 '18 at 2:46

Hartogs' theorem is provable without Replacement. The trick is to note that the isomorphism with ordinals is really unnecessary.

Instead you want to just look at well-orders modulo order isomorphisms. In fact, one can look at $\{X\subseteq\mathcal P(A)\mid (X,\subsetneq)\text{ is well-ordered}\}/\cong$, or in other words, look at the set of chains of subsets of $A$ which are well-ordered by $\subseteq$, modulo the order-isomorphism relation.

It is not hard to show that this set is both well-ordered, and does not embed into $A$. So if $A=\omega$, the resulting order type is uncountable by definition.

Note that Replacement was never used here. We never mapped each well-order to its order type as a von Neumann ordinal. We just looked at sets of sets of chains of subsets, with a bunch of Separation.

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    $\begingroup$ It might also be worth pointing out, that replacement is not needed to construct the ordering of the equivalence classes; since we can take $E_0 \le E_1$, iff, for every $a\in E_0$, there is some $b\in E_1$, with $a\subset b$. $\endgroup$ – Not Mike Dec 4 '18 at 2:08
  • $\begingroup$ It is intuitively obvious why the set that you suggest is well-ordered under the order that @NotMike given in his comment but, in my work to prove it, I can't get nothing. Can you give me a hint? $\endgroup$ – Gödel Dec 4 '18 at 22:49
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    $\begingroup$ @Gödel: Given two well-orders, one is isomorphic to an initial segment of the other. $\endgroup$ – Asaf Karagila Dec 5 '18 at 0:27

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