צפה בגרסה המלאה : שאלה בתורת הקבוצות

21-11-2008, 19:05
אני קורא קצת על קריפטוגרפיה מספר אלקטרוני אחד, ורשום בו:

Considering that already as children we learned counting as something to be taken for granted and that we were readily convinced of such facts as that two plus two equals four, we must turn to surprisingly abstract thought constructs to derive the theoretical justification for such assertions. For example, set theory allows us to derive the existence and arithmetic of the natural numbers from (almost) nothing. This "almost nothing" is the empty (or null) set ø := { }, that is, the set that has no elements. If we consider the empty set to correspond to the number 0, then we are able to construct additional sets as follows. The successor 0+ of 0 is associated with the set 0+ := { 0 } = { ø }, which contains a single element, namely the null set. We give the successor of 0 the name 1, and for this set as well we can determine a successor, namely 1+ := { ø, { ø }}. The successor of 1, which contains 0 and 1 as its elements, is given the name 2. The sets thus constructed, which we have rashly given the names 0, 1, and 2, we identify—not surprisingly—with the well-known natural numbers 0, 1, and 2.

לא בדיוק הבנתי את זה, ואשמח אם תסבירו לי על זה מעט.
הבנתי שהאות היוונית ø מסמלת קבוצה ריקה, כלומר קבוצה חסרת איברים, שלה קוראים קבוצת null.
לאחר מכן לא הבנתי. |:
תוכלו להבהיר לי מעט את העניינים בבקשה?
תודה רבה. :)

בהמשך רשום כך:

This minimal and thus uniquely determined successor set N_IMAGE_DISPLAY_HERE is called the set of natural numbers, in which we expressly include zero as an element.

כלומר, 0 הוא חלק מקבוצת המספרים הטבעיים? כך שכל ההוכחות באינדוקציה נכונות גם עבור n=0?