Operationes primae nvmerorvm natvralivm et relationes ordinis

Svmmarivm

Haec lectio additionem, mvltiplicationem et potentiam nvmerorvm natvralivm ex svccessore axiomatvm Peani definit, adhibito theoremate recvrsionis, qvod praestat definitiones recvrsivas vnam tantvm operationem determinare; eivs existentiam admittit, vnicitatem per indvctionem demonstrat, et proprietates operationvm dedvcit: commvtativitatem, associativitatem, distribvtivitatem et cancellationem. Deinde relationem ordinis per additionem definit, cvm proprietatibvs trichotomiae, transitivitatis et monotoniae, et operationes inversas, svbtractionem et divisionem, vna cvm condicionibvs qvibvs exsistvnt. Principivm indvctionis instrvmentvm est omnivm demonstrationvm.

Proposita discendi

Theorema recvrsionis envntiare et eo definitiones recvrsivas additionis, mvltiplicationis et potentiae nvmerorvm natvralivm scribere.
Proprietates operationvm ex earvm definitionibvs per indvctionem demonstrare.
Ordinem nvmerorvm natvralivm per additionem definire eivsqve proprietates trichotomiae, transitivitatis et monotoniae adhibere.
Agnoscere qvando svbtractio et divisio dvorvm nvmerorvm natvralivm exsistant, et cvm eis operari non egrediendo ex nvmeris natvralibvs.

Additio

Operationes nvmerorvm natvralivm iam a schola notae svnt; hic agitvr vt accvrate mathematice fvndentvr. In lectione de axiomatibvs Peani nvmeri natvrales definiti svnt ex $1$ et ex operatione svccessoris, qvae cviqve nvmero natvrali $n$ seqventem attribvit, $S(n)$ . Additio ex svccessore dvabvs regvlis definitvr: $1$ addere est svccessorem svmere, et svccessorem ipsivs $m$ addere est svccessorem svmmae cvm $m$ svmere.

n + 1 = S(n), \qquad n + S(m) = S(n + m)

Definitio hvivs generis recvrsiva appellatvr: prima regvla valorem pro $1$ statvit, secvnda qvemqve casvm ad priorem redvcit. Iam vero dvas tales regvlas re vera operationem definire dvas affirmationes distinctas continet: exsistere fvnctionem qvae eis pro omnibvs nvmeris natvralibvs satisfaciat (existentia), nec dvas fvnctiones diversas esse qvae eis satisfaciant (vnicitas). Ambae theorema recvrsionis efficivnt, qvod Dedekindo debetvr: hic cvrsvs existentiam admittit, vt post tabellam exponitvr, et vnicitatem demonstrat. Forma generali envntiatvr, vt etiam mvltiplicationi, potentiae et definitionibvs recvrsivis lectionvm seqventivm inserviat.

Factvm admissvm (theorema recvrsionis). Sint $X$ copia et $c$ elementvm ipsivs $X$ , et ponamvs datam esse, pro qvoqve nvmero natvrali $n$ , regvlam $g_n$ qvae cviqve elemento ipsivs $X$ elementvm ipsivs $X$ attribvit. Tvm exsistit fvnctio $f$ ex $\mathbb{N}$ in $X$ talis vt $f(1) = c$ et $f(S(n)) = g_n(f(n))$ pro omni nvmero natvrali $n$ , eaqve fvnctio vnica est.

Existentia in hoc cvrsv sine demonstratione admittitvr. Eivs demonstratio postvlat vt fvnctio $f$ tamqvam copia parivm constrvatvr: minima copia parivm qvae $(1, c)$ continet et qvae, cvm qvoqve pari $(n, x)$ , etiam $(S(n), g_n(x))$ continet; ac deinde vt per indvctionem confirmetvr cviqve nvmero natvrali in ea vnvm tantvm altervm elementvm respondere. Haec constrvctio ad theoriam copiarvm pertinet nihilqve arithmeticae qvae seqvitvr confert. Vnicitas avtem conseqventia directa est qvinti axiomatis Peani.

LemmaVnicitas fvnctionis per recvrsionem definitae

Si dvae fvnctiones $f$ et $h$ ex $\mathbb{N}$ in $X$ dvabvs condicionibvs theorematis recvrsionis satisfacivnt, aeqvales svnt. Figvra valores vtrivsqve in primis nvmeris natvralibvs repraesentat.

Demonstratio

$\dato{A}{A = \{ n \in \mathbb{N} : f(n) = h(n) \}}$
Sit $A$ copia nvmerorvm natvralivm in qvibvs dvae fvnctiones congrvvnt. Satis est ostendere $A$ copiam omnivm nvmerorvm natvralivm esse.
$f(1) = \resaltar{c} = h(1)$
Nvmervs $1$ ad $A$ pertinet, qvia ambae fvnctiones valorem $c$ in $1$ habent.
$f(S(k)) = g_k(\resaltar{f(k)}) = g_k(\resaltar{h(k)}) = h(S(k))$
Si $k$ ad $A$ pertinet, etiam $S(k)$ : ambae fvnctiones eandem regvlam $g_k$ eidem valori adhibent.
$A = \mathbb{N}, \quad (\forall n \in \mathbb{N})\,(f(n) = h(n))$
Per qvintvm axioma Peani, $A$ copia omnivm nvmerorvm natvralivm est: dvae fvnctiones aeqvales svnt.

Additio ex theoremate obtinetvr primo addendo $n$ fixo, svmptis $X = \mathbb{N}$ , $c = S(n)$ et, pro omni $m$ , regvla $g_m$ qvae cviqve nvmero natvrali svvm svccessorem attribvit: fvnctio $f$ qvam theorema praebet ea est qvae cviqve $m$ nvmervm $n + m$ attribvit, et dvae regvlae additionis dvae condiciones theorematis svnt. Eodem modo, cvm $c = n$ et regvla qvae $n$ addit, mvltiplicatio obtinetvr, et cvm $c = b$ et regvla qvae per $b$ mvltiplicat, potentia. $m$ addere deniqve idem est ac $m$ vicibvs svccessorem adhibere.

ExemplvmDvo et dvo qvattvor svnt

Notissima arithmeticae affirmatio theorema est qvod dvabvs regvlis additionis et nominibvs $2 = S(1)$ , $3 = S(2)$ et $4 = S(3)$ demonstratvr.

Demonstratio

$2 + 2 = 2 + S(1)$
Altervm addendvm vt svccessorem ipsivs $1$ scribimvs.
$2 + S(1) = S(2 + 1)$
Per secvndam regvlam, svccessorem addere est svccessorem svmmae svmere.
$S(2 + 1) = S(S(2))$
Per primam regvlam, $1$ addere est svccessorem svmere.
$S(S(2)) = S(3) = 4$
Cvm $S(2) = 3$ et $S(3) = 4$ , eventvs obtinetvr.

Ex regvlis additionis eivs proprietates per indvctionem dedvcvntvr. Pro omnibvs nvmeris natvralibvs $a$ , $b$ et $c$ :

Commvtativitas: $a + b = b + a$ .
Associativitas: $(a + b) + c = a + (b + c)$ . Qvam ob rem $a + b + c$ sine parenthesibvs scribi potest.
Cancellatio: si $a + c = b + c$ , tvm $a = b$ .

Commvtativitas in problemate solvto 5 lectionis de axiomatibvs Peani demonstrata est, et associativitas ibi vnvm ex problematibvs propositis est. Cancellatio in problemate solvto 1 hvivs lectionis demonstratvr.