document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=424968>Question 424968</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that 1+1 is equal to 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #296100 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=296100\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "A few simple definitions in Peano terms. 1 is the successor of 0, or S0. 2 is the successor of 1, or so on\r<BR>\n" );
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document.write( "Any equation is equal to itself, so we can start with S0 + S0 = S0 + S0 (1 + 1 = 1 + 1)\r<BR>\n" );
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document.write( "Any expression of the form A + SB is equivalent to SA + B, so S0 + S0 is equivalent to SS0 + 0. Thusly we can prove that (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+1+=+2+%2B+0 ALIGN=MIDDLE  ALT=\"1+%2B+1+=+2+%2B+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Since the expressions X + 0 and X are of equal value (by the Peano definition of addition,) SS0 + 0 can be reduced to SS0. Thus, we have proved that 1 + 1 = 2 according to Peano's axioms. (I think... hopefully I haven't skipped over anything or taken an illegal shortcut. It's been a while since I've worked with Peano.)\r<BR>\n" );
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document.write( "Algebraically, we can approach the problem somewhat simpler:\r<BR>\n" );
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document.write( "Take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+1+=+2 ALIGN=MIDDLE  ALT=\"1+%2B+1+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides, and you see that statement is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+1+-+1+=+2+-+1 ALIGN=MIDDLE  ALT=\"1+%2B+1+-+1+=+2+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplifying both sides, you can get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+1 ALIGN=MIDDLE  ALT=\"1+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is pretty self-evident.\r<BR>\n" );
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