Prove that in Peano arithmetic, 0 is the only number with the property that it is not the successor of another number.

trying to get your homework done?:P

or you could have just said natural numbers and quoted:
"1. Zero is a number.

2. If a is a number, the successor of a is a number.

3. zero is not the successor of a number.

4. Two numbers of which the successors are equal are themselves equal.

5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S. "

besides, if your read
http://www.encyclopedia69.com/...etic/peano-arithmetic.htm (see the proof of 1 + 1 ) = 2 and using axiom 3
you can prove your statement quite readily:P

I"m not doing any more.

My fiancee and I are looking at getting married in a place in St Mary's county (amongst other places in Maryland and Washington (the real one))

martian - is the natural numbers the only set which satisfies the peano axioms?

I wasn't asking whether or not 0 has the property of not being the successor of any number, but if it necessarily had to be the only number with that property. Axiom 3 merely states that 0 has that property, but does not specify that no other number can have that property.

i want to hear more about detmers wedding plans, personally..

I think that is about it so far. Instead of wedding cake I want to have a gigantic meatloaf, but otherwise nothing has really been worked out.

Although I went to college there, I'm not too familiar with possible places to get married there. There's really like nothing at all down there except for a military base and a college.

http://www.seentvcanada.com/perfect-meatloaf-p-682.html

it might be too small for a wedding meatloaf, but it would be PERFECT!

Rockman, yeah I know. But if you use axiom 3 and something similar to the proof listed in my second link (bottom paragraph) you can show that this is the case.

And as for it being the only set: I would say almost any countable set with either a finite lower or a finite upper bound (ie a definition of "0") containing countably infinite unique elements would qualify. for example you could take the sequence {0,1/n) and define the order as 0, 1/1, 1/2,.. where (1/(n+1)) is the successor of ((1/n)). and 1/1 is the successor of 0... I think:p

Since the natural numbers are not the only set, then specifying natural numbers and quoting those axioms would have been a different question. Because its fairly obvious that in the natural numbers, 0 is the only number which is not a successor. You can also find numerous sets other than the natural numbers which satisfy the peano axioms AND have 0 as the only number that is not a successor.

But its not as obvious (and possibly not even true), that in all systems which satisfy the peano axioms, that 0 is the only number which is not a successor.

Oh, I thought you lived there.




braden... that does look PERFECT! Fortunately it is too small. I do wonder how I would even bake a meatloaf that big... would definitely have to cook for a long time... definitely have to cover that so the edges don't get too burnt... probably need to have a lot of water in there to help keep things moist...

Nope, I live in Montgomery county now. I did live in Charles County for a bit after college, but never in St. Mary's County.

Rockman: I would think that by the induction principle every other element (other than 0) would have to be a successor.
other than the case where you have the set {0, 0, 0, 0, ...)

Intuitively I would think that it's something like defining a vector space in linear algebra where 0 has to be part of the vector space and 0 is unique.

Would need to think about this more

"Set theory is a disease from which mathematics
will one day recover" (Poincare)

martian - every element other than 0 is a successor is true if 0 and its successors are the entire set. But what if 0 and its successors are not the entire set?

The way you phrased the induction axiom seems to imply that, but the way I've seen the induction axiom phrased does not seem to imply that. I guess there is variance in the way the induction axiom is phrased, but it seems odd for such a large variance to exist.

additionally, 0 cannot be the successor of itself, because 0 is not the successor of any number. So {0,0,0,0...} is not valid.

Rockman, I guess phrasing is important.A quick google search on Peano gives a bunch of different links with multiple phrasings of the axiom. Maybe if there was something shown mathematically for that axiom it might be clearer what was intended. The original source is probably the best bet:P
and you are correct about{0,0,0,...}.

Yep, it seems important that the induction axiom implies that 0 is the only number that is not a successor, or for the 3rd axiom to explicitly state that. Otherwise, I think there are some odd systems that could be properly described as systems of peano arithmetic.

So the easiest way to resolve it is just to phrase the induction axiom properly :)

Peano started at 1

the answer is eleventy one

I licked a Peano once...


*Runs Away*

once?

I'm with braden on this one.

How big of a wedding are you trying to have Detmer?

There are 88 keys on a standard piano.

Right now we have our list and it has 80 people on it. (So pretty much family and wedding party with dates). We also plan to have parties in Maryland, Wisconsin and Washington which will be open to a broader base of friends, but the wedding itself it will be 80 tops.

Detmer: You wouldn't cook the meatloaf cake as one huge cake. You'd need to section it off and make it with smaller sections. For cooking, make sure the cake is a little drier so that it holds together better (more breadcrumbs). After its cooked and cooled some on a draining rack, you'd need to level off the tops. Use ketchup inbetween the loaves, but make sure to leave about 1 inch of clearance room so the ketchup doesn't get on the sides. After you have the cake stacked, you'll want to trim the outside to give it a good appearance. For the "frosting" buy some instant potatoes and put on a very small layer around the "cake". After that you can proceed to frost the whole cake with the potatoes. For decoration, you can pipe the potatoes just like frosting to give it a cake look. If you need to add any colors, food coloring can be added with the potatoes and it will be fine.

I love how this started as a math question and ended in meatloaf cake decoration lol. Way to steal the thread detmer lol

I am well aware the best way to do this is with layers. That is a good tip on the amount of clearance to leave for the ketchup (barbecue sauce in my case, since I hate ketchup (yes, I am aware that barbecue sauce is mostly ketchup... this is purely empirical hatred)). The mashed potato frosting is absolutely ingenious. Do you have any thoughts on reheating, since clearly meatloaf is best served warm? Baked mashed potatoes are fine, but there might be issues keeping with the time necessary to reheat the whole cake? Or do you think this has to be something done like during the ceremony? I suppose that even the cake could be made first then reheated shortly before hand and frosted last.

After you frost it, you can chill it overnight. Once chilled, you should be able to cut it much easier. After its cut, then it could be microwaved and served. I'm not sure where about you are having your reception, but if there is a kitchen at the facility then perhaps this is doable. If not, then as you stated: Perhaps cook it, chill it, slice it, heat it frost it.

Good luck and take pictures!

Rockman: "Prove that in Peano arithmetic, 0 is the only number with the property that it is not the successor of another number."
Assume the opposite and assume that there are two sequences. What happens in the end is that the only way this follows the axioms is that if both sequences are equal.

However if I stretch your definitions a bit to assume elements in a set then I can do the following:
lets take a set in R2 like (0,1),(0,2),(0,3)....
where (0,1) acts like "0"

Then I can take another set. (1,0),(2,0),...
where (1,0) = acts like "0"
If I take the union of both sets then I have disproved the above. However I suspect that this is not a Peano set.

Martian - that depends on the wording of the induction axiom.

If you take the union of those two sets, you must actually identify a specific element as "0". You can't say that two elements 'act like 0', you have to designate a specific element as 0 to satisfy the axiom that 0 is a number.

Once that is the case, then according to the wording of the induction axiom that you first posted, then 0, and the set of all its successors must be the entirety of all numbers. Therefore, if you choose just (0,1) as representing 0, then the set of (0,1),(0,2),(0,3).... would be the entirety of all numbers, and (1,0) would not be a number because it is not in that set.

If the axiom of induction is worded in a way that implies that every number must have 0 as its ancestor if you go back far enough, then it directly implies that 0 has to be the only number that is not a successor of another number.

Rockman: that's very true. AFAIK most algebras and set theory require zero to be unique.
What one needs is a more mathematically precise and consistent definition of what a peano set actually is, which we appear to lack.

If the axiom of induction is not worded in a way to prohibit
(0,0), (0,1), (0,2) ... U (1,0), (1,1), (1,2) .... being a peano set, then it actually wouldn't prohibit the union of all sets of the form (x,0), (x,1), (x,2) ... where x is a real number from being a peano set, which would result in a peano set with a cardinality not equal to aleph null.

That would be an interesting implication, which I think makes it clear that 0 has to be unique, therefore the wording of the axiom of induction MUST force 0 to be unique (because the other 4 axioms do not force 0 to be unique).

well
even if we back up one step
consider
0,1,2,3,4....
and 0,0.5,1.5,2.5,3.5...
each of those is a peano set and the union between the two is a peano set.
In fact I could conceivably union a countably infinite number of peano sets together and still have a peano set.
I think the difference is that things change when you aren't doing a union on a finite number of sets.. then you need to be careful. The mathematical theory on that kind of thing gets rather messy.