# Are the following size rules taken from the herediarily finite world consistent with ZFC?

Let $$x^*$$ be the set of all cardinalities that are strictly larger than the cardinality of $$x$$ and strictly smaller than the cardinality of $$P(x)$$. Formally, that is:

Define: $$s=x^* \iff \forall y (y \in s \leftrightarrow \exists k (|x| < |k| < |P(x)| \wedge y=|k|))$$

Are the following rules consistent with ZFC? if so, what's their consistency strength?

$$\forall x (|x|>2 \to |x| < |x^*| <|P(x)| )$$

$$\forall x (|x|>2 \to ||x|^{+*}| > |P(x)|)$$

$$\forall x \forall y (|x| > |y| \wedge |y|>2 \to |x^*| > |P(y)|)$$

Where $$|x|^+$$ is the successor cardinal of the cardinality of $$x$$.

Where cardinality $$||"$$ is defined in the customary manner after Von Neumann's.

Also for better reading the above rules can be written in terms of cardinals as:

$$\forall \kappa \, (\kappa >2 \to \kappa < |\kappa^*| < 2^\kappa )$$

$$\forall \kappa \,(\kappa >2 \to |(\kappa^+)^*| > 2^\kappa)$$

$$\forall \kappa \,\forall \lambda\, (\kappa > \lambda >2 \to |\kappa^*| > 2^\lambda)$$

Where $$<",>"$$ are the known cardinal inequalities; and $$\kappa^*=\{\lambda\mid \kappa < \lambda < 2^\kappa \}$$

• The "set of all cardinalities that are..." means "any set with cardinality that..."? If yes, the first is vacuously true. – Martín-Blas Pérez Pinilla Apr 15 at 8:09
• @Martín-BlasPérezPinilla, let me give an example, take $x$ to be a set whose cardinality is 3, then x* would be the set {4,5,6,7}, so x* is the set of all cardinalities that are strictly larger than 3 and strictly smaller than 8. – Zuhair Apr 15 at 8:11
• Maybe relevant :math.stackexchange.com/questions/1483349/…. – Martín-Blas Pérez Pinilla Apr 15 at 10:31
• If your theory is consistent, it has quite high consistency strength as it implies a massive failure of SCH: If $\kappa$ is a strong limit then $2^\kappa\geq\aleph_{\kappa^+}$. If it is inconsistent, the singular cardinals are the troublemakers. Using Eastons theorem, one can construct a model where your theory holds on the regular cardinals. Your theory implies that all strong limit cardinals are $\aleph$-fixed points, which prevents one from applying e.g. Galvin-Hajnal to get an inconsistency. – Andreas Lietz Apr 15 at 13:31
• By the way, maybe more people would consider your question if it was phrased more conveniently. For example, your third axiom already implies the second one. Moreover this is only about cardinal arithmetic, so why not talk directly about cardinals instead of the cumbersome notation $\vert x\vert$? – Andreas Lietz Apr 15 at 13:32