Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

138

138 ~~~~CARDINAL ARITHMETIC[PRII

[PART III

*115-18. I-.t'Prod/c =t'K [*83-81]
*115-2. I- K ic CIS3 arithm. Ki, S',c e Cls2 exci [(*115-02)]
a,8 /3 K. p E a. a (7 /.3 2! O A% a-. p P =
[*115-2. *84-11]
*115-211. F:KeCls~arithm.a,/3EK.pfa.o-E/3.3!pAa.:).a=/3
Dern.
[*13-13].p 6aeaA3.
[*115-21].a =/3::)I. Prop
*115-22. I-:. K eCls~'arithin.:) S""K eClS2excl a,/3EK.3[! s'anAs'/3. )a,,,,.a=/3
Dem.
-. *4011. ) I-:H! s'a A s'3.=. (x,p,a(). p Ea * 7e/3. x ep. xca-.
[*10-35] u(po1e~-/.!~ 1
F.(1).*115-211.)D
I-:.Hp.):a,/3eKc.3!s'ans'/3.).a=/3. (2)
[*30-37]:). S'a = s'/38 (3)
F. (2). (3). *8411 F. Prop
Observe that, although" ""K E Cls2exci" follows from
"(a,8 /3 K. 3! s'a A s'/3.:),. a = /3"
the converse implication does not hold. If there were two different classes
a and /3 having the same sum, we might have 3! s'a A s'/38, i.e. 3! s'la, without
having a = /3, in spite Of " S"K E C1s2 excl." In proofs, less use can be made
Of 1cS((K,6CJS2excl " than of "a,/3eK.!S'a ns'/3 a, a=/3." If A'1E K or
tfAe-.JE,6K, the latter implies S rK l*l
*115-23. F:. KE6ClS3 aritbm. ):
Prod"'K e CIS2 excl: a,8 /3 K.3!, Prod'a A Prod'/3.. a = /3
Dem.
F. *83-62. )I- e Prod'a nProd'/3. )uC s'a ns'/3 (1)
F. (1).*24-58. ) F: isProd'aAnProd'/3.3!Y.)D.3g! s'a ^s'/3 (2)
F. (2). *115-22.:)F:Hp.a,/3e6K. weProd'a mProd'/3.3[! w.)D. a =/ (3)
F.*83-16. Transp. DF:A eProd'a AProd'3. D. a = A./3=A (4)
F.(3).(4).:)F:. llp.)a,/3e6K.3[! Prod"'aA Prod'/3. ). a =/3. (5)
[*30-37]. Prod'1a=Prod'/3 (6)
F.(5).Q36). *84-11. ) F. Prop
*115-24. F:KeCCls3 arithm.=*.et K, ErS'K ECls -+1 [*115-2. *84-14]
*115-25. F:K eClssarithm. ).E4'K Cl-+1. E'S'K Cl1 --- 1 [*84-3. *115-2]