# New PDF release: An Introduction to Set Theory

By W. Weiss

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Y Φ(x, f, y)], which simply states that we have a well defined procedure given by Φ. Let us now go to the general context in which the above example will be a special case. For any formula Φ(x, f, y, w) of the language of set theory, we denote by REC(Φ, N, w) the class { x, y : (∃n ∈ N)(∃f )[f : n → V ∧ f (x) = y ∧ ∀m ∈ n Φ(m, f |m, f (m), w)]}. 36 CHAPTER 4. THE NATURAL NUMBERS We will show, under the appropriate hypothesis, that REC(Φ, N, w) is the unique function on N which satisfies the procedure given by Φ.

We can also carry out recursive definitions on ON. This process is called transfinite recursion. For any formula Φ(x, f, y, w) of the language of set theory, we denote by REC(Φ, ON, w) the class { x, y : (∃n ∈ ON)(∃f )[f : n → V∧f (x) = y∧∀m ∈ n Φ(m, f |m, f (m), w)]}. Transfinite recursion is justified by the following theorem scheme. For each formula Φ(x, f, y, w) of the language of set theory we have: Theorem 14. y Φ(x, f, y, w)]. Then, letting F denote REC(Φ, ON, w), we have: 45 1. F : ON → V; 2.

0 + α = α; 2. If β < δ then α + β < α + δ; 3. α + sup S = sup {α + η : η ∈ S}; 4. α + (β + δ) = (α + β) + δ; 5. If α < β then α + δ ≤ β + δ; 6. 0 · α = 0; 7. 1 · α = α; 8. If 0 < α and β < δ then α · β < α · δ; 9. α · sup S = sup {α · η : η ∈ S}; 10. α · (β + δ) = (α · β) + (α · δ); 11. α · (β · δ) = (α · β) · δ; 12. If α < β then α · δ ≤ β · δ; 13. 1α = 1; 14. If 1 < α and β < δ then αβ < αδ ; 15. αsup S = sup {αη : η ∈ S}; 16. α(β+δ) = αβ · αδ ; 17. (αβ )δ = αβ·δ ; and, 18. If α < β then αδ ≤ β δ .