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My question is inspired by the fact that there is no universal set (at least in ZF). There are many abstract objects such as group, ring, field, vector space, topology, etc. such that we can say about embedding. In a sense, we can interpret $A$ embeds $B$ as $A$ 'contains' $B$. So it seems that there is no such universal objects (universal group/ring/field...) that embeds every possible objects (group/ring/field ..). How can I prove/disprove it?

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    What exactly do you mean: do you want a foo (group/ring/field...) x such that every other foo is a sub-foo of x or that every other foo is an element of x?2011-08-25
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    @Ilmari Karonen Thanks for grammar correction! I'm still a novice in English so it helps me a lot.2011-08-25
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    @Miha Yes. If anything makes you incomprehensible feel free to modify my question to make it easier to understand.2011-08-25
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    There is in fact a ‘universal’ ring, but it's not quite the thing you suggest. Rather, it is a ring object in the classifying topos of the theory of rings, and every ring in every topos is the image of the universal ring under a unique geometric morphism.2011-08-25
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    If you are working in ZFC, remember that every infinite set can be made into a ring. In particular, if $A$ was a universal ring, how would you embed $P(A)$ into it?2011-08-25
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    every finite group is a subgroup of $\cup S_n$2011-08-25

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There are cardinality limits in these cases, too. That is, there exist groups rings fields topologies (etc.) of arbitrarily large cardinal. So no " universal" ones exist.

But for example there are nice things like: a universal separable metric space: that is, a separable metric space such that any other separable metric space is isometric to a subset of it. Or a universal countable group. And so on.

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    See e.g. http://en.wikipedia.org/wiki/Urysohn_universal_space2011-08-25