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$\omega+1\sim\omega$ but $\omega+1\not\simeq\omega$

$\sim$ is defined as follows:

$A\preceq B$, if there is an injection from $A$ to $B$ and

$A\sim B$, if $A\preceq B$ and $B\preceq A$

Or can you explicitly construct an isomorphism $1+\omega\simeq w$ which fails for $\omega +1$

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    what part are you stuck on? – 2017-01-13
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    @William understanding the difference between $\omega +1$ and $1+\omega$ – 2017-01-13
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    @ketum I don't understand your comment. You say you're stuck on understanding the difference between $\omega+1$ and $1+\omega$ but there's nothing about $1+\omega$ in your question. – 2017-01-14

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In ordinal addition, $a+b$ means you line up $a$ things and put $b$ things after them. If you take $\omega = (0,1,2,3,4,\ldots)$ then $1+\omega=(c,0,1,2,3,4,\ldots)$ and $\omega+1=(0,1,2,3,4,\ldots,c)$ Note that $\omega+1$ has a last element while $1+\omega$ does not. Doesn't $1+\omega$ look like $\omega$? That should suggest an order isomorphism between them. If you try to make an order isomorphism between $\omega+1$ and $\omega$, where can $c$ go?